Math is a tool. You can hardly blame a saw when you fail to sharpen your knife with it.

So is math necessarily a poor tool? In my view, it's a wonderful language through which concepts can be expressed concisely and unambiguously. It's quite powerful to express an economic problem as a math problem because oftwn this means the problem is already solved in full generality (e.g. Recognizing an incapacitated lot problem as an integer programming problem which has easy algorithms for solution).

I've heard some complaints that the temptation to pretend a problem is accurately modeled by a solved math problem is quite strong - this is probably one of the issues you're getting at. Again, nothing wrong with using a tool its proper place. The fault is unambiguously with the user.


Is there any statistics available on forecasting failures?

Like what percentage of financial/economical forecast fail?

Well it depends on what you call a forecast.

Considering that there is a large uncountable number of people who are called (or call themselves) financial,economic and policy analysts all over the world who together make what one would think millions of forecasts made everyday, there is probably a number which fail and a number which succeed.

So where are the numbers on it?

I would suspect no general statistic exists because its measurement does not have a practical use from a policy perspective. I say this in the sense, that it is a concern exclusively of private businesses and companies to care how well their forecasts work. providing statistics on their forecasting success would be exclusively for marketing purposes.

  Or how often do politicians create bad policies based on wrong forecasts?

This part of the question I recommend for Politics SE as it is  more political than economic.

Hope this helps


What is the interest rate that on the vertical axis of money markets?

The thing to keep in mind is that these diagrams refer to simple teaching economic models. These models do not necessarily reflect the complexity of the real world.

  The money market diagram showing liquidity preference and money supply
  intersect at an equilibrium level of interest rate, but what interest
  rate is it?

Within the teaching model, there is a portfolio allocation between "money" and some interest-paying instrument (labelled "bonds" or "bills"). The interest rate is the interest rate paid on that hypothetical instrument.

The usual interpretation is that this the short-term "risk-free" rate of interest. (Risk-free implies is that they are assumed to be free of default risk.) In the United States (for example), there are a number of such instruments: Treasury bills/bonds, the fed funds rate (domestic interbank rate), OIS, repo, interest-bearing reserves, etc. These instruments have interest rates that are generally close to each other.

Simplified models that consist of a pair of intersecting lines will not be able to deal with the fact that such instruments have a variety of maturity dates (from overnight to 30 years).

The general assumption when discussing these models is that private sector borrowing rates will remain close to this risk-free curve. The fact that this is not always true is another part of reality they cannot deal with. That said, credit spreads for high quality borrowers tends to be less volatile than the level of the risk-free rate, and so the risk-free rate does roughly capture the general tendency of interest rates.

  Is it the base rate (I presume this is the rate at which commercial
  banks lend out loans to borrowers)

The only "base rate" I can think of is a rate associated with central bank dealings with banks. The prime rate is a benchmark borrowing rate set by banks for dealing with customers. For example, people have loan interest rates that contractually are set at "prime + x%".

  FED Funds Rate/LIBOR?

FED Funds represents reserve trading between domestic banks; it is considered a risk-free rate (no credit premium).

LIBOR is a rate for unsecured lending between international banks (in London). It has a credit risk premium embedded in it.


Even after the comments exchange I am not totally sure what is the problem of the OP, but I will answer the question in the title.

Assume that we have a model

$$Y_i = \beta X_i + u_i,\;\;\; i=1,...,n$$

Assume that we obtain a "random sample" of observations from $\{Y_i, X_i\}$. This makes each observation $i$ identically and independently distributed from all other observations.

Now consider the random variable

$$v_i=(X_i-\mu _X)u_i = (X_i-\mu _X)(Y_i - \beta X_i) = X_iY_i - \beta X_i^2 - \mu_XY_i+\mu_x\beta X_i$$

while also

$$j\neq i : v_j=(X_j-\mu _X)u_j = (X_j-\mu _X)(Y_j - \beta X_j)=X_jY_j - \beta X_j^2 - \mu_XY_j+\mu_x\beta X_j$$

$v_i$ is a function of random variables $\{Y_i, X_i\}$ only, while $v_j$ is a function of random variables $\{Y_j, X_j\}$ only which are independent from  $\{Y_i, X_i\}$. So  $v_i$ is independent from $v_j$.

More over, since $X_i$ follows the same distribution as $X_j$, and $X_j$ follows the same distribution as $Y_j$, we have that $X_iY_i$ follows the same distribution as $X_jY_j$.Etc. So we also conclude that $v_i$ will have the same distribution as $v_j$.


Autocorrelation function of a random walk process

By recursive substitution you obtain $y_t = \sum_{j=0}^t e_{t-j}.$ Thus $\forall p \in \mathbb{N} \hspace{.2cm} y_{t-p} = \sum_{j=0}^{t-p} e_{t-p-j}.$

Under the usual white noise assumption for the errors $i \neq j \Rightarrow E(e_je_i) =0\Rightarrow Cov(y_t, y_{t-p}) = E(\sum_{j=0}^{t-p} e_{t-p-j}^2) = (t-p+1)\sigma_e^2.$

$Var(y_t) = E(\sum_{j=0}^{t} e_{t}^2) = (t+1)\sigma_e^2 \hspace{.2cm} \wedge Var(y_{t-p}) = E(\sum_{j=0}^{t-p} e_{t-p-j}^2)(t-p+1)\sigma_e^2.$

$\Rightarrow \rho_t(p) = \frac{(t-p+1)\sigma_e^2}{\sqrt{(t+1)\sigma_e^2}\sqrt{(t-p+1)\sigma_e^2}} = \frac{\sqrt{(t-p+1)}}{\sqrt{(t+1)}}.$

$\Rightarrow lim_{p\rightarrow 0} \rho_t(p) = 1 \hspace{.2cm} \wedge \Rightarrow lim_{p\rightarrow t} \rho_t(p) =  \frac{1}{\sqrt{(t+1)}}$.

$\Rightarrow$ for fixed $p$ $lim_{t\rightarrow \infty} \rho_t(p) = lim_{t\rightarrow \infty} \frac{\sqrt{(t-p+1)}}{\sqrt{(t+1)}}= 1$


How is bartering (goods for goods) conceptually related to trading (goods for money) and monetary (money for money) exchange?

Consider an economy where there are $N$ goods and assets. How do you purchase a good? If $n \cdot (n-1)$ forms of exchange are possible because all $n$ goods can be all other $n-1$ goods then this is a barter economy. If there is only a single good that can be exchanged for all other goods then only $n-1$ forms of exchange are possible then this is a monetary economy where that single good is "money". See, for example,  Kiyotaki and Wright (1993). Used this way, Trade Equilibrium $\supset$ Barter Equilibrium $\cup$ Monetary Equilibrium. A no trade equilibrium is sometimes called autarky.


How much to earn to break-even throughout life

Generally, only the top 1% pay significant taxes. Nearly half of taxes in the US are paid by the top 1% of taxpayers.

So in order to pay an average tax burden you need to make hundreds of K a year.


Does consumer theory assume that utility functions are consistent overtime?

In The Economic Approach to Human Behavior, Gary Becker said:

  The combined assumptions of maximizing behavior, market equilibrium, and stable preferences, used relentlessly and unflinchingly, form the heart of the economic approach as I see it.

The emphasis is mine. Here "stable preferences" refer to preferences (and by association, the utility functions representing them) that are more or less the same across different periods. The reason for preference stability is obvious. If we allow preference or utility function to change arbitrarily, then we'd be able to explain pretty much anything by attributing the cause to some appropriately chosen change in people's preference.

When it comes to the intertemporal preferences in particular, I agree with @MaartenPunt that, at least in principle, one can incorporate time-dependence into utility function. For instance, in the usual discounted utility framework, we have
\begin{equation}
U(\mathbf x_t)=\sum_{t=0}^\infty D(t)u(\mathbf x_t)
\end{equation}
where $\mathbf x_t$ is a vector of consumption goods at time $t$, $D(\cdot)$  is a discount function (e.g. $D(t)=\delta^t$ as in an exponential discounting model), and $u(\cdot)$ is the time-invariant period utility function. To incorporate time-dependence, we can simply allow $u(\cdot)$ to also be a function of time
\begin{equation}
U(\mathbf x_t)=\sum_{t=0}^\infty D(t)u(\mathbf x_t,\color{red}t).
\end{equation}
To make @MaartenPunt's third comment more explicit, suppose
\begin{equation}
u(\mathbf x,t)=\alpha_t^1v(x^1)+\cdots+\alpha_t^iv(x^i)+\cdots+\alpha_t^nv(x^n)
\end{equation}
where $x^i$ denotes the quantity consumed of good $i$ and $\alpha_t^i$s are the time-dependent weights on the utility derived from each good $i$.
So the same consumption bundle will generate possibly different levels of utility  in different time periods. For example, $\alpha_{10}^\text{candy}>\alpha_{40}^\text{candy}$ would capture the fact that a 10-year-old values a candy more than a 40-year-old does. On the other hand, a time-invariant preference would imply that $\alpha_t^i=\alpha^i$ for all $t=0,1,\dots$.

The above discussion is however distinctly different from the paper you linked to in the comment, which is about dynamic (in)consistency of choices. In the literature on intertemporal choices, the main focus is usually about whether some optimal consumption profile decided at time $t$ will remain optimal when reevaluated at some future time $t+k$.
Usually papers in this literature maintain the assumption of time-invariant period utility function, i.e. $u(\mathbf x,t)=u(\mathbf x)$, but play with various forms of the discount function $D(\cdot)$ (e.g. hyperbolic or quasi-hyperbolic discounting) to generate predictions that match experimental data.


Total differentiation in a IS curve context

Total differentiation means you are differentiating all variables in the expression. So if we start with
$$y = C\{[y + B - T(y + B)]; [M + \frac{B}{r}]\} + I(r) + G$$
we get:
$$ d y = C_y^{'} dy + C_y^{'} dB - C_y^{'} dTdy - C_y^{'} dTdB + C_A^{'} dM + C_A^{'} \left( \frac{dBr-Bdr}{r^2} \right) + I'dr + dG $$
Here you are using just basic calculus such as chain rule accordding to which $\frac{d}{dx}[y(u(x))] = \frac{dy}{du} \frac{du}{dy}$, plus you use the rule that with total differentiation  $dy(x_1,x_2) = y_1' dx_1 +y_2'dx_2$ and also quotient rule $\frac{d}{dx}[F(x)/G(x)]= \frac{F'G(x) - F(x)G'}{G(x)^2}$.
Next you just move everything to the LHS of the equation to get
$$dy - C_y^{'} dy - C_y^{'} dB + C_y^{'} T'dy + C_y^{'} T'dB - C_A^{'} dM - C_A^{'} \left( \frac{dBr-Bdr}{r^2} \right) - I'dr - dG = 0$$
Now just collect terms and clean up the mess above and you directly get:
$$(1 + C_y^{'}(1 -  T'))dy - (I_r - C_A \frac{B}{r^2})dr - [C_y (1 - T') + \frac{C_A}{r}]dB- C_A^{'} dM  - dG = 0$$
The expression looks difficult because it has a lot of terms but you should not get intimidated by its look.


What are the advantage of chain-linked prices over constant prices?

The advantage of chain linking is that component weightings drift away from their weight in nominal GDP as the reference period recedes.

This is a St. Louis Fed document on the methodology: link to website.

I am not familiar enough with the details to summarise it better than that document.


What is the effect of redirecting government tax intake to a consumer retirement saving incentive?

If you've ever looked at the ratchet effects model, you might be able to glean insight from there? Basically, if a worker is paid in a piece rate $a + bq = S, for \space e(q)$, where e is the effort exerted by the worker, we know that it is the most efficient--normally--to set $b = 1$ (no deadweight loss of "taxation" of labor, so to speak), assuming that e is measurable. However, given the cost of effort, the firm will always try to guess what that ($C(e)$) is so that $a + be - C(e) = 0$, to minimize the salary $S$ that is paid out. Workers, anticipating this, ratchet their productivity downwards, since working hard will tell a firm that the job being done by the worker is very easy, and not worth much pay.

So this means the optimum is for the firm to set $b > 1$ in a two period model where you observe effort, and then set the piece rate pay in the second period. Imagine how this might work for employers. They know that the amount they put into the Kiwisave is dependent on how much they pay out to the worker, given some chance that the employee actually puts aside the 3%. So the firm will set pay lower than they normally would if the government raised their investment into Kiwisave. The question is, would firms lower $b$ or $a$. When you optimize a simple model like this, $a$ usually doesn't go into the FOC, so probably what will happen is $b < 1$.

I think that increased government spending wouldn't necessarily increase employer costs if more employees opt in. If employers only had to pay the 2% if employees successfully paid their share that is, 100% chance employees who opt in, then more government spending would do almost nothing, barring elasticity. Those who opt in would simply have their wage set lower by the firm.

All in theory.


Aggregated demand of households given utility function

You just need to use the condition $$ MRS_{q_{1j},q_{2j}} = \frac{p_{1}}{p_{2}} $$ to obtain $$ 3 \frac{q_{2j}}{q_{1j}} = \frac{p_{1}}{p_{2}} \;\;\;\; \text{(1)}$$ Then solving for $p_{1}$ and plugging this into the budget constraint you obtain: $$ y = 3p_{2} \frac{q_{2j}}{q_{1j}}q_{1j} - p_{2}q_{2j}$$ $$ \Rightarrow q_{2j} = \frac{y/4}{p_{2}} $$ Accordingly, solving for $p_{2}$ from condition (1) and plugging this into the budget constraint yields: $$ q_{1j} = \frac{3y/4}{p_{1}} $$.

From this you can obtain your function for the aggregated demand for goods $q_{1j},q_{2j}$, as a function of the prices.


Can a country's default on internal debt affect the rates of the external debt?

Probably, but it's not visible because a country is usually already in shambles before a default, and data collection for domestic debt crises is poor.

Here is a list of defaults:

http://www.nber.org/papers/w13946

Here's some rate data:

http://www.statistikbanken.dk/statbank5a/SelectVarVal/saveselections.asp

http://www.tradingeconomics.com/argentina/inflation-cpi

https://en.m.wikipedia.org/wiki/Historical_exchange_rates_of_Argentine_currency

There have been many cases where countries default on only internal debt. Argentina did this. I can't find bond data anywhere. But when it defaulted on domestic debt (1989), inflation went up and the currency fell. Which is pretty much the same as an increase in external interest rates.


Homothetic function versus straight Engle curve

A general result using indirect utility functions, is that expenditure linear in income ("straight-line" Engel curves) is produced if and only if the indirect utility function has the "Gorman-polar form".

The original paper is

Gorman, W. M. (1961). On a class of preference fields. Metroeconomica, 13(2), 53-56.

...where we read that a linear Engel curve is produced when we have a homothetic utility function (Gorman does not use the word, he writes "homogeneous" and in a footnote, enhances that to "function of a homogeneous function", which is how a homothetic function is defined).

But the reverse does not hold:homotheticity is sufficient for linear Engel curves, but not necessary.

For example, linear Engel curves are also produced by a utility function that reflects "quasi-linear" preferences, $U(x,m) = u(x) + m$, which are not homothetic.

A homothetic function is properly defined mathematically as follows:

Let $f(\mathbf x)$ be a homogeneous function of some degree, and let $g$ be a function with non zero derivative. Then $g[f(\mathbf x)]$ is called a homothetic function. In economics, we usually require for the derivative of $g$ to be strictly positive, so that we remain inside economically meaningful territory.


There are a couple of good papers on this that I know of.

Keane, Michael and Richard Rogerson, “Micro and Macro Labor Sup-
ply Elasticities: A Reassessment of Conventional Wisdom,” Journal of
Economic Literature, 2012, 50, 464-476.

"The response of aggregate labor supply to various changes in the economic environment
is central to many economic issues, especially the optimal design of tax policies.
Conventional wisdom based on studies in the 1980s and 1990s has long held that the
analysis of micro data leads one to conclude that aggregate labor supply elasticities
are quite small. In this paper we argue that this conventional wisdom does not hold
up to empirically reasonable and relevant extensions of simple life cycle models that
served as the basis for these conclusions. In particular, we show that several pieces
of conventional wisdom fail in the presence of human capital accumulation or labor
supply decisions that allow for adjustment along both the extensive and intensive
margin. We conclude that previous estimates of small labor supply elasticities based
on micro data are fully consistent with large aggregate labor supply elasticities."

Peterman, William B, “Reconciling Micro and Macro Estimates of the
Frisch Labor Supply Elasticity: A Sensitivity Analysis,” Technical Report,
Federal Reserve Board of Governors 2014.

"Microeconometric estimates of the Frisch labor supply (0 to 0.5) tend to be much lower than the values used by macroeconomists to calibrate general equilibrium models (2-4). This paper explores whether the gap in these ranges can be explained by two restrictions present in the micro Frisch elasticity that are implicitly relaxed in these values of the macro Frisch elasticity. First, the micro estimates focuses only on prime-aged married males who are the head of their household, while the macro values incorporates the whole population. Second, the micro estimates only include fluctuations in hours on the intensive margin, while the macro values also incorporate fluctuations on the extensive margin. Within a consistent microeconometric estimation strategy, this paper estimates a micro Frisch elasticity of 0.2, and, upon relaxing the two restrictions, a macro Frisch elasticity of about 3. The increase in the estimates suggests that these two restrictions can explain the gap. However, given that this paper demonstrates that the estimates of the macro Frisch elasticity are fairly sensitive to the estimation procedure, the sample of agents, and also the margins of fluctuations that are used for estimation one must careful choose the appropriate value for calibration purposes."


How to show that GLS estimator is consistent in regression model?

If the regressors are strictly exogenous to the error term (which is the initial standard assumption made), meaning, if the error vector is mean-independent of the regressor matrix

$$E(\mathbf u \mid \mathbf X) = \mathbf 0$$

then consistency follows immediately. Set for convenience $\mathbf \Omega ^{-1/2} \equiv \mathbf C$. Then

$$\text{plim}\left(n^{-1} (\mathbf C\mathbf X)'\mathbf u \right)= \text{plim}\left(n^{-1}\mathbf X'\mathbf C'\mathbf u\right)$$

This is a $k \times 1$ matrix. What would a typical element will be, say the first?

The matrix $\mathbf C'\mathbf u$ is a $n\times 1$ vector,

$$\mathbf C'\mathbf u = \left(\sum_{i=1}^nc_{1i}u_i,..., \sum_{i=1}^nc_{ni}u_i\right)'$$

Mutliplied by the first row of $\mathbf X' = (x_{11}\;...\;x_{1n})$ gives

$$\left [\mathbf X'\mathbf C'\mathbf u\right]_1 =x_{11}\cdot \sum_{i=1}^nc_{1i}u_i\;+...\;+ x_{1n}\cdot\sum_{i=1}^nc_{ni}u_i$$

Now strict exogeneity comes into play: it is stronger and so implies that

$$E(\mathbf u \mid \mathbf X) = \mathbf 0 \implies E(x_{1i}u_j)=0\;\;\forall i,j$$

So the above sum of sums, scaled by $n^{-1}$ will tend to zero asymptotically, since all products in each sum will tend to a zero expected value. This proves consistency.

Note that strict exogeneity is really needed, for the case where the matrix $\Omega$ is a function of the regressors. In such a case, we would be looking at products of the form

$$x_{1i}\cdot h(\mathbf x) \cdot u_j$$

which would still go to zero in expected value terms, because mean-independence implies orthogonality between the error term and any function of the regressors.

If $\Omega$ does not depend on the regressors, then we could weaken strict exogeneity to the RHS condition, which is  stronger than "contemporaneous uncorrelatedness", the latter needed for OLS.

I  leave the case for asymptotic normality to anyone interested.


How could having more US dollars solidified the 1991 Argentine peg to the US dollar?

Simply having more dollars may have served two purposes. Firstly, more dollars would mean that the central bank could have bought more pesos and exerted some upward pressure on the exchange rate. Secondly, fixed exchange rates can fall because of expectations about the central bank's credibility to its peg. If the prevailing expectation is that the central bank does not have the resources (reserves) or the will to defend the peg, pesos will begin to be sold en masse in the expectation of a devaluation.


Consequences to lending and value of national currency from a negative interest rate and 140-year mortgages in Sweden

There is some concern about the interest rates (currently at -0.5%) fueling a housing bubble in Sweden. This article at Fidelity states:

  In a bid to track the ECB, Sweden has cut its interest rate below
  zero, a radical move that involves charging banks to hold some types
  of deposits with the aim of encouraging them to lend.

  Similar trends have emerged in other small, open economies, such as
  those of Denmark and Switzerland, which also have subzero rates.

  In Sweden, the central bank governor has warned that the low rates
  could be creating a housing price bubble.

A reason why this risk exists in particular for Sweden is because the regulation has allowed the borrower to pay only the interest on common mortgage types, and since the interest rates have been very low for many years, the sizes of these payments have been low.

These mortgages are referred to as "Swedish style mortgages" in this report published by the Swedish Central Bank (dated April 2015).

Overall the situation has stimulated borrowing and fueled the growth of the overall indebtedness of the households. Above report states on p. 4 that:

  Sweden, in particular, has seen household indebtedness rise from 90%
  of disposable income in 1995 to 172% in 2014

Current situation is still as described as on page 14 in the Sweden - Selected Issues paper published by the IMF (dated August 2014), shown in below image.

On p.15 this IMF report states that:

  If progress under the current voluntary approach remains limited, a binding maximum amortization period for new mortgages would be
  helpful. Swedish banks are working to raise amortization through
  voluntary means. However, the picture remains largely unchanged. The
  share of new mortgages granted in 2013 that has an amortization plan
  has remained broadly constant from 2012. And, of the existing mortgage
  stock, a significant 40 percent of households either increased or did
  not decrease their debt in 2013. For the remaining 60 percent of
  households, it would take 100 years to fully pay down their debt if
  they continued to amortize at current speeds.

In fact they attempted to regulate the mortgages last year but according to this article it looks like the proposed changes were rejected by an appeals court. The article states that

  Under the original plans, customers would have been asked to pay two
  percent of the value of their mortgage every year until they had
  repaid 30 percent of the loan. They would then have been required to
  pay at least one percent a year until they hit the 50-percent mark.

  But the watchdog backtracked after an administrative court of appeal
  in Jönköping in southern Sweden suggested that the proposed changes
  were not supported by Swedish law.

I am not sure however if they were able to pass that law later in some modified form.

Understandably the current level of indebtedness poses a large risk for the banks in the event of a downturn on the housing markets. However the already existing debt levels (as discussed in this report) in comparison to disposable income may make it difficult to implement drastic changes to the regulations.

The reason for the interest-rate cut you asked about seems to have been to avoid further deflation. One of the consequences of this has been, according to this article, an overall increase in house prices. So while at least two of the symptoms of a housing bubble (high prices and high levels of household debt) are present in Sweden, there are also factors that play against a US-style mortgage crisis, such as:

The ability of households to make interest-only mortgage payments
without penalty for an extended period.
Large amounts of recent immigrants (refugees from mainly the middle
east) that will require housing. In addition to this Sweden already
had low levels of new construction even before the latest influx of
immigrants.
The relatively small size of the country; people tend to not move
geographically as far away as in USA for reasons such as employment and may stay at a particular address longer.

Update: According to this article, Sweden did change the maximum length of a mortgage to 105 years earlier this year.


How can I draw an edgeworth box with 3 different goods?

I assume what you're asking based on your comments is: "How can I visualize indifference curves for 3 goods?" I can think of three options:

1) Use a tool like Matlab, or its open-source equivalent, Octave, to plot 3 dimensional indifference curves. Here is a tutorial on how to do that.

2) Make a series of 2-dimensional indifference curves for two of the goods, and fix a different level of the third good for each one, that is, plot a series of cross-sections of the 3 dimensional box.

3) Using paper and pencil, draw a general $n$ dimensional Edgeworth box, then, set $n = 3$.


Why is the Marginal Utility of losses deminishing in Prospect Theory?

The value function used in Kahneman's prospect theory (which your plot shows) is supposed to capture empirically observed behavior of people's attitudes towards gains/losses as well as to risks in those domains.

In the domain for gains, people are usually risk averse. However, in the domain for losses, people do tend to take larger risks. A nice illustration is the observation that when a gambler is losing money (relative to the amount he came to the casino with), he tends to wager on higher stakes (larger risks), perhaps in an attempt to "win back" the losses.

So the convex curvature in the domain of losses is meant to capture this type of behavior.

In fact, prospect theory's treatment of people's risk attitude is even more nuanced than the above. In addition to the S-shaped value function you showed, it also has a component that allows for non-linear probability weighting. Together, the value function and non-linear probability weighting generate the following four-fold classification of risk attitudes:
\begin{array}{c|cc}
&\text{gains}&\text{losses}\\\hline
\text{low probability}&\text{risk loving}&\text{risk averse}\\
\text{high probability}&\text{risk averse}&\text{risk loving}
\end{array}

See Kahneman and Tversky (1992) for more detail.


The term "Most Favoured Nation" is misleading to me. Please explain

You seem to have it the wrong way around.

The act of country A extending Most Favoured Nation status to country B only affects the import tariffs on country B's products, not those on other countries' products. MFN status means that the tariffs on the Most Favoured Nation's products cannot be higher than the lowest import tariff on similar products from other countries to country A. (Hence the name 'most favoured', as it will have the lowest tariff.) Similar products from other countries without MFN status from country A may have higher tariffs than those imposed on country B's products.

There are some caveats. E.g. if country A is in a free trade zone with countries X and Y and charges no tariffs at all on their products then even if country A grants country B Most Favoured Nation status, it does not mean it cannot impose any tariffs against country B's products. Basically country B has to get the lowest tariff, but free trade zones are not regarded as having 0 tariffs, rather as being exempt from tariffs.


Hayek's The Constitution of Liberty is in my opinion the place to start. It has a lot of references to John Stuart Mill (which whom he strongly disagrees, btw), and puts forward his argument on freedom. There is also a very recent (but rather long) analysis of Hayek's book (available online here), which dissects his influences.


Why do we spread knowledge across nations?

This is a long comment that is too long for the comment box.
In fact, many research projects in US are secretive, especially the ones related to military. At the same time, much, much more research projects in China are secretive, including some agricultural ones, which are not military-related but considered as national security-related by Chinese government.
There has always been a China-threat theory in US, and a US-threat theory in China. The majority of US military are nationalists which could be considered as right-wing, while the majority of US academia are left-wing globalists who think China is less threatening. So the later care more about free-market and free knowledge while the former might want to put some restrictions on China.
By the way, there is a culture difference on "world order" between the East and the West. For thousands of years, the Western world is comprised of hundreds of smaller yet independent states, while almost all of the Far East nations could have been a tributary subject of the "Center Empire". People from the "Center Empire" could naturally believe that there will be only one major leader in the world, while the westerners might believe that the world is consist of many equal entities. The latter is of course less concern about knowledge-sharing.
The globalists believe that the global challenges faced by all human are more threatening than China. This could be one of reasons why the globalists have gained more support in the special year of 2020.


What would be the economic impact of regulating advertising?

The first thing we must ask is about which part of advertising are we speaking? Are we talking about advertising on television and other entertainment mediums or are we talking about advertisement via the packaging of the good itself?

One impact either way would be an increase in competition amongst monopolistically competitive firms. These firms are often selling products which accomplish the same ends, however they differentiate their goods in order to give them a competitive edge. The main way which they convey this differentiation to consumers is through advertising. Therefore, if they cannot advertise this differentiation, consumer information may be limited and they may not see a difference between the firm's product and other products in the market. This will lead to parity among products and will move the market closer to a purely competitive market.

Now, sometimes the only differentiation may be the advertising itself (in the form of packaging). For goods such as water bottles (barring whether the water is enhanced somehow) this is the case. There has been much research done to ascertain the best way to package your products in order to draw the consumer to the product. If this was limited, then markets such as these would move towards a purely competitive market.


Trade-off between determenistic and random alternatives in a one-shot game

Let's assume that the player is an expected utility maximizer, and that the utility they get from a payoff of $\$z$ is $u(z)$, for any $z$ in $\mathbb{R}$ such that $u$ is strictly increasing.

You are essentially asking about certainty equivalents. Call $\sigma$ the gamble that pays $y$ with probability $\frac{1}{2}$ and $0$ with probability $\frac{1}{2}$. Then, the utility of the $\sigma$ (under the EU hypothesis) is $U(\sigma) = \frac{1}{2}u(y) + \frac{1}{2}u(0)$. This means that the player is indifferent between this gamble and getting $u^{-1}(\frac{1}{2}u(y) + \frac{1}{2}u(0)) = u^{-1}(U(\sigma))$ for sure. This is called the certainty equivalent of $\sigma$. Notice, $U$ transforms a gamble over monetary payoffs into utilities, and $u^{-1}$ transforms the utilities back into monetary payoffs. If $x$ is a constant gamble then $u^{-1}(U(x)) = u^{-1}(u(x)) = x$.

If $x$ is larger than the certainty equivalent of $\sigma$, the player will choose $x$. You are asking the inverse (if $CE: \sigma \mapsto u^{-1}(U(\sigma))$ you are interested in $CE^{-1}$). Nonetheless the underlying mechanics are the same, once the utility function has been specified. The $y$ you are interested in satisfies

$$ y = u^{-1}(2u(x) - u(0)).$$

The deeper question here, however, is how is the certainty equivalent related to the gamble itself. In general (for a very ugly $u$), not much can be said. However, it is common practice to make some structural assumptions on $u$ to answer the questions you are asking. For example, what if we assume the utility index is differentiable and concave, for example $u(x) = ln(x)$. Then it is relatively straightforward to show that $y(x)$ is increasing and convex (loosely speaking, the relative difference between $x$ and $y$ is increasing, as suggested in your post, although it does not capture the complete reluctance to choose the risky prospect in the third example (although this can be captured by other, more complicated models)).

It is worth noting, all of this assumes EU, which while extensively used in applications, especially game theory, is not without its empirical failures. There are many other "psychological" models which might provide a different resolution to your inquiry.


Rule for Number of Strategies in Bayesian Games

Consider a type space $\mathcal T=\{1,2,\dots,T\}$ and an action space $\mathcal A=\{1,\dots,A\}$. With $A=T=2$, you correctly found that there are $S=4$ different pure strategies mapping $\mathcal T \to \mathcal A$. Now fix $A$ and add one more type, so $T=3, A=2$. Type 1 and 2 still have the same number of different actions (4), and the strategy can be completed by two different actions by type 3. That is, $S= 4\cdot2=8$. If you added a fourth type, these $8$ combinations can again be completed to a full strategy with 2 different actions, i.e., $S=16$... and so on such that $S=2^T$.
Next, go back to $A=T=2$, fix $T$ and add actions. You see that type 1 has $A$ actions that can each be paired with $A$ actions by type 2, giving you $S=A^2$.
Combining your insights you find $S=A^T$. In simultaneous-move games, $\mathcal A$ is just the set of simultaneous actions. In more general games, $\mathcal A$ contains all complete plans of actions for each contingency in the game.


How can a "producer face a limited market as regards sales and yet a highly competitive market as regards price"?

Could someone explain and also provide concrete examples of how it might be possible that a "producer faces a limited market as regards sales and yet a highly competitive market as regards price"?

I believe an example would the Bertrand duopoly. In Bertrand duopoly you have only two firms so all sales are made just by two firms, but we know that in Bertrand duopoly, thanks to competition, the optimum price will be exactly equal to marginal costs like in perfect competition.
For example, assume that there two firms competing on price. Their demand is given by:
$
q_1=  D_1(p_1,p_2) =
\begin{cases}
 D_1(p_1)/2, \text{ if } p_1 =p_2 \\
 D_1(p_1), \text{ if } p_1<p_2\\
      0, \text{ if } p_1>p_2
\end{cases}
$
the marginal costs of will be given by $MC$. In such case the firm's best responses will be given as in the picture below (taken from wikipedia). As can be seen from the picture the best responses intersect exactly where prices equal marginal costs so in such situation you get prices that are equivalent to perfectly competitive prices, even though the sales are made on duopoly market.


What is the difference between Stochastic game and Bayesian game ?

In a Bayesian game, information is incomplete. To cope, players have beliefs about the state of the game. In a sense, each player strategizes as if the game was as he or she believes. So each player operates in his or her own world. And if every player plays a Nash equilibrium in one's own world, that's a Bayesian Nash equilibrium.

In a stochastic game, the information about the current state of the game may indeed be public. At a given time step, the next state is determined by the current state, the strategy profile played at that time step, and some stochastic process (like a Markov chain, for example).

We can have a stochastic Bayesian game where information is incomplete and there is a stochastic switching device like a Markov chain.


Connection between saving and investments-Y=C+I+G+NX

The $I$ is actually not just capital spending. In some models that might be true, for example, in Solow growth model we will assume that all investment is capital spending but $I$ is not just that.
For example, $I$ would include both purchases of new houses (although argument could be made they are capital goods, they would not satisfy more narrower definitions requiring that capital being factor of production), or investment in inventory for example (see more detailed explanation in Blanchard et al. Macroeconomics: a European Perspective, pp 41-43).
Also, spending on capital good is literary considered saving in economics. In economics saving has more broad meaning than what ordinary people just consider 'saving'. For example, in economics when you buy stocks you are also saving even if you spend money doing so.  In economics by definition the portion of income that is not consumed is saving (see ibid. Macroeconomics: a European Perspective pp 569-570).
PS: As pointed in the comments, you should have minus in front of $C$ so the equation should be: $ Y-G-C = I+NX$.


why in some countries trade deficits for a few years generates economical crisis and in United States 40 years of deficit hasnt?

Persistent trade or balance of payments deficits generate debt and other liabilities, either in the currency of the importer or in some foreign currency.  This may be sustainable so long as lenders expect to be repaid in something of value.

The United States borrows in dollars, and the general expectation among its creditors is that its debts will be honoured (there was some doubt in 2011 and 2013) and that others will accept dollars in the future.

For Argentina, this expectation is much weaker.  In the past there have been defaults on borrowing in dollars (eight times since 1824, most recently in 2001 and 2014).  Alternatively, Argentina could borrow in pesos, but there have been frequent bouts of hyperinflation, which makes foreign lenders reluctant to lend.

So for the United States, its historical behaviour has created a degree of trust among its creditors which allows a sustained deficit.  Argentina's history makes it harder to borrow, so any new suggestion of a persistent deficit means it soon finds financing imports more difficult, thus creating a crisis.


IS-LM , Eurozone and Interest Rate Parity

It’s the same currency and interest rate structure for all countries. The only difference is that there is a default premium for some countries. You see the exact same thing in Canada, with the Canadian provinces trading at different yields.
The ECB is literally the union of all the national central banks. When the ECB makes a decision, it is implemented at the national central bank level.

The IS/LM model has severe limitations. The ECB sets interest rates; it’s ability to control “the money supply” (which aggregate?) is debatable. (It can increase the size of the monetary base, but that does not imply that bank deposits etc. will grow.)


What is a good handbook for behavioral economics?

Sanjit Dhami's The Foundations of Behavioral Economic Analysis is arguably the most comprehensive review of the work in behavioral economics to date.


With whom does the US have the largest deficit using the trade in value-added (TiVA) methodology?

The OECD provides value-added trade statistics from 2005-15. The table below, compiled by the Federal Reserve Bank of St. Louis, shows the U.S. trade balance in 2015 for several major trading partners in terms of gross trade and value-added trade.
Using the trade in value-added methodology, the U.S. trade deficit with China is reduced by 13 percent but still the largest.

Another figure that shows the U.S. trade balance from 2005-15 with several major trading partners in terms of real gross trade and real value-added trade.


You're probably confusing two things: the objective function $pf(x) - wx$ being used in profit maximization problem: $\max pf(x) - wx$, the objective function is concave in $x$ and thus, you can solve the maximization problem. However, a profit function $\pi(p,w)$ being a result of maximization problem identifies the maximum profit given level of prices and wages $\pi^*(p,w)$ (and it's convex in $p$ and $w$ as you said).


Why's Bernie Sanders wrong to allege that JP Morgan received more than $390 billion in financial assistance from the Fed?

Example 1. If I repeatedly make 365 one-day \$1 loans to you over the course of 365 days, then the unadjusted loans (or "financial assistance") I make to you are \$365.

However, in effect I've simply lent you \$1 for one year. So, the term-adjusted (or we could say year-adjusted) loans I've made to you are only \$1.

Example 2. Loan A: I make you a one-day \$365 loan.

Loan B: I make you a one-year \$1 loan.

Both Loans A and B are \$1 term-adjusted loans.

But note that they are not obviously exactly equivalent. Most would consider Loan A a bigger favor than Loan B. This though is debatable. But in any case, the point here is that nobody would consider these two Loans to be exactly equivalent.

This is explained in the cited document, paragraph above Table 9 and again in the Note to Table 9. An example is also given on pp. 130–1:

  Table 8 aggregates total dollar transaction amounts by adding the total dollar amount of all loans but does not adjust these amounts to reflect differences across programs in the term over which loans were outstanding. For example, an overnight PDCF loan of \$10 billion that was renewed daily at the same level for 30 business days would result in an aggregate amount borrowed of \$300 billion although the institution, in effect, borrowed only \$10 billion over 30 days. In contrast, a TAF loan of \$10 billion extended over a 1-month period would appear as \$10 billion. As a result, the total transaction amounts shown in table 8 for PDCF are not directly comparable to the total transaction amounts shown for TAF and other programs that made loans for periods longer than overnight.

The cited Sanders quote is not obviously "stupid" though, as the Reddit commenter suggests. Example:

Say in an emergency, the Fed has to lend \$10T to Bank A for just one day. By the above procedure, we'd translate into to a term- or year-adjusted loan of \$10T ÷ 365 ≈ \$27.4B.

Say the Fed also lends \$27.4B to Bank B for one year.

It's not clear that the degree of assistance the Fed provided to both banks is equivalent. Most would say that the Fed did Bank A a bigger favor.


First a disclaimer: Trade imbalances are something that we should expect to happen, especially between two massive economies like the US and China. In fact, it will probably be very hard to find a pair of countries that don't have a trade imbalance (unless they don't trade at all).

This does not mean that there is something wrong necessarily, however, a lot of the discussion seems to think that tariffs are solving a problem or making it worse when trade balances are not necessarily positive or negative.

Now, more directly to your point, the trade balance depends on many things. To list a few: the economic growth of both countries, their relative efficiency on tradable goods, the terms of trade, the relative exchange rates, and whether they are controlled or not, the tariffs or restrictions to trade, the relative purchasing power, the real interest rates in each country, etc. I think that the US and China are very different countries, so it will be quite heroic to try to explain everything with a single variable like tariffs. The trade imbalance is certainly  the result of multiple economic conditions.

Sure enough, if you want to reduce how much you import from a country, tariffs will help, but the real question is why would you want to do this?


We don't know.  Consider the following possibilities:

The tariff lasts long enough that the United States soybean farmers switch from soybeans to some other crop.  When the tariff is dropped, they don't change back.  Brazil keeps the business.
Brazilian farmers open new land and buy equipment specifically for soybeans.  When the tariff ends, Brazil and the US split the business in some fashion (implicitly via market actions).  Some farmers in both countries may go out of business or switch crops as a result.
Brazilian soybeans that had shipped to Europe are now shipped to China.  Europeans buy soybeans from the US that they had bought from Brazil.  Transport costs may climb, but both the US and Brazil continue to produce soybeans at about the same rate as they did prior to the tariff.  When the tariff goes away, business patterns may go back to the previous pattern.
The US uses soybeans to make biofuels or otherwise finds new ways to use soybeans.
Chinese consumers might replace some soybeans with some other food in their diet.  Maybe they decide they like the new food better and they never eat as many soybeans as they did previously.
Producers and consumers engage in long term deals.  So once the market is changed, it stays changed even after the impetus for change is gone.

We might know how much of the soybean market is dominated by long term deals, but other than that, we don't really know how things will change.  It's quite possible that all these things happen to some level.

When something like this happens, it may make both permanent and temporary changes.  Someone might be able to work out a model whereby we can speculate as to how much of the change would be permanent and how much would be temporary.  But any such model is going to be somewhat of a guess.


What impact would introducing 15 times more soy worldwide do to the price of soy?

Roberts and Schlenker (2013) estimate the price elasticity of demand for agricultural commodities to be about -0.08 to -0.05. These are elasticities estimated over much smaller changes in $p$ and $q$ than you are suggesting. Remember that the elasticity is effectively percent change in $q$ for a percent change in $p$, so we can ask what the percent change in $p$ is for a percent change in $q$.

If production is 15x more soy than there is currently (in total, so globally) then this is a 1500% ((1*15+1 - 1)/1) increase in soy. Using the high end (most elastic), we get that prices have to fall 18,750% (15/-0.08) to clear the market at this new higher quantity. This is impossible, so this tells us that our local elasticity measures won't do to solve this problem. It also suggests that there would be no reason to add so much soy bean production, because demand is likely unable to handle so much soy.


Elasticity of Substitution of CRS Production Function

I am not sure if it is intuitive but this is because because CRS function is homogenous of degree 1.
Full derivation:
First, general formula for any arbitrary elasticity of substitution between $L$ and $K$* is given by (see Sydsæter et al. EMEA pp 430):
$$\sigma_{L,K}  = \frac{-F_K'F_L'(xF_K'+ yF_L')}{xy \left( (F_L')^2F_{KK}^{''} - 2 F_K'F_L'F_{KL}'' + (F_K')^2F_{LL}''\right)}, \text{ for: } F(K,L)=c \tag{1}$$
Where c is arbitrary constant. This is our starting point.
Now by definition property of CRS production function is that they are homogenous of degree 1 (since by definition we have CRS when $F(tK,tL)= tF(K,L)$).
If $F$ is homogenous of degree 1, then the numerator of (1) will be $= −F_K'F_L'F$.
This is because of the Euler's theorem which tells us that if:
$$f(x,y) \text{ is homogenous of degree k} \implies xf_x'(x,y) +  yf_y'(x,y)= kf(x,y)$$
Next Euler theorem also implies that (assuming $f$ is twice continuously differentiable) that:
$$xf_{xx}''(x,y) +  yf_{yx}''(x,y)= (k-1)f_x'(x,y)$$
$$xf_{yx}''(x,y) +  yf_{yy}''(x,y)= (k-1)f_y'(x,y)$$
The above implies that in our case:
$$K F_{KK}'' = - LF_{KL}'' $$ and that $$L F_{LL}'' = - K F_{LK}'' = - K F_{KL}'' $$. Hence the denominator will be given by:
$$ -F{KL}''\left( L^2 (F_L')^2 + 2KLF_K'F_L'+ K^2(F_K')^2\right)= - F_{KL}''(KF_K'+ L F_L')^2 = - F_{KL}''F^2$$
We use Euler theorem above again.
Now finally we are done:
$$\sigma_{LK} = \frac{−F_K'F_L'F}{- F_{KL}''F^2} =\frac{F_K'F_L'}{ F_{KL}''F}  $$
I personally do not think the result above is very intuitive (if it is so its intuition eludes me), but it is a consequence of CRS function being homogenous of degree 1 and this result actually applies for any elasticity of substitution between two variables of arbitrary function that is homogenous of degree 1 and twice continuously differentiable.

 * or $y$ and $x$ for that matter this generalizes to any elasticity of substitution problem


Help in interpreting this graph? Assets/Liability ratio

Wouldn't it be better to use equity instead of assets for a measure of
  leverage?

No, but the correct ratio is

$$\frac{\text{Assets} }{ \text{Equity}}$$

Reasoning: with "leverage" we want to obtain information on how much a company uses debt for financing. And we want to have the correspondence "higher leverage $\to$ more debt financing".

Moreover, in order to be consistent with the definition of "leverage" ("the exertion of force by means of a lever"), we want the information on debt to be in the denominator, since we want to form a ratio.

In order to achieve all these together, we need to put equity on the denominator, so that lower equity (implying higher debt for same level of assets) increases the value of the ratio, and we get the higher leverage.


In 21st century and a globalized world, is economic growth still the metric by which to measure a state's success?

Depends on what you mean by "success".

Economic growth is simply change in gross output expressed in market prices. Prices tell you what citizens want and how much of it they have. Equating marginal utility with price is a useful abstraction in this case.

Economic growth is trivially desirable: it's about people making things they need.

When "success" is defined in some progressive terms, it's no longer trivially desirable. Zero employment policies — like those in the Soviet Union — hurt productivity. Premature investments in clean tech worsen intertemporal tradeoffs. You actually get back to GDP to measure these successes.

Still, measurements of economic growth can get better. Like Stiglitz et al's suggestion to use net disposable income instead of GDP per capita (see Commission on the Measurement of Economic Performance and Social Progress, Mismeasuring Our Lives). Adjusting per-capita metrics for inequality would also be a good idea.


Why is 10% the necessary upper bound for a negative interest rate?

Assuming your home is safe, would you keep your money in a bank that only gave 96% of it back? (-4% interest rate)

What if your home has mice which eat exactly 10% of your money? Would you then keep your money in the bank that only returns 96%?

And finally, what if every bill has a 10% chance of being eaten? This means that in expected value you lose 10% of your money.

In the above situations, what would be the lowest interest rate you would accept from the bank?


Why would governments sell bonds when the economy is in recovery?

The article pretty much answers this:

China has issued 100 billion yuan (about 14.1 billion U.S. dollars) of special government bonds for COVID-19 control measures in a bid to support local infrastructure construction and epidemic prevention and control, the senior official added.
It is set to counter the downward pressure on economic growth, expand tax and fee reduction policies to help enterprises tide over hard times, and strengthen budget balance to mitigate the adverse impact of the epidemic on fiscal revenue growth.

Even if the economy is doing better, it is expected to otherwise be below their growth targets.
(If you have a more specific question, you should re-phrase your question accordingly. If you want to find someone with expertise on Chinese government policy making (not me!), I am unsure how many read this website.)


I will only provide a general outline here since this is clearly a homework question.

You know that you should have, at an optimal point, that the marginal utility of each of your two goods is equal and so you can use:

$$MU_x=MU_y$$

You also have a budget constraint, yes? something like:

$$x+y=P_x(initial endowment of x) + P_y(initial endowment of y) + Income$$

Now you have enough equations to solve, I think. Isolate either x or y using your budget constraint. To be concrete here, I will assume you isolate x.

Take this representation of x and plug it in for x where you've equated your marginal utilities. This should allow you to solve for y in terms of exogenous factors. Take this representation of y and plug it back into $MU_x$. This gives both x and y in terms of exogenous factors.

So, all you are doing is equating marginal utilities and also using the budget constraint to express the optimal bundle (optimal amounts of goods x,y) in terms of exogenous factors.


Beyond Michel De Vroey's book, I know of no other book on the history of macroeconomics, as it is praised by macroeconomists. See for example this review.

De Vroey has a true vision of macroeconomics, he shares it with his reader and gives clear guidelines to understand the developments in the field.

Anyway, you will find below some nice references that could be useful:

The State of Macro, by Olivier Blanchard of MIT.
What do we know about Macroeconomics that Fisher and Wicksell did not? , by Olivier Blanchard of MIT.
The Macroeconomist as Scientist and Engineer, by Gregory Mankiw of Harvard.
Slideshares on a Brief History of Macroeconomics by Markus Brunnermeier and Delwin Olivan of Princeton.


In consumer theory, preferences are called rational if they are transitive and complete. Some nomenclature: An alternative A is weakly preferred to another alternative B, if you either like A more than B or you are indifferent between the two. Transitive means that whenever an alternative A is weakly preferred to alternative B, and B is weakly preferred to C, then A is weakly preferred to C. Complete means that for any two elements A and B of the set of possible alternatives, either A is weakly preferred to B, B is weakly preferred to A, or both. In short, preferences are rational when they are defined for all possible alternatives, and when they don't contradict each other in obvious ways.

Rational expectations are a different beast and are specific to different models. Agents in a model have rational expectations if they use the model to form expectations about the future. This means that expectations are consistent with the model/theory in question, and it also means that every rational expectations model has a different "theory" about how people form their expectations. Rational expectations are as much a concept to solve dynamic models as they are an assertion on people's behavior. They are commonly used because they make sure that people are not systematically wrong about their beliefs, and they keep models' properties from being driven by more arbitrary assumptions on how people form expectations.

I don't think there is a lot of thought going into alternatives to rational preferences and most people are content with making the assumption that preferences are rational. There is a big literature in macro and finance on relaxing rational expectations assumptions. A lot of people work with models of bounded rationality, in which agents use a simpler approximation to the true model to form expectations.


Diminishing returns and economies of scale

In production with more than one input, "diminishing returns" refers to what happens when we increase one input while keeping all the rest constant. "Economies of scale" is a more informal term for "increasing returns to scale" and so relates to what happens when we increase all inputs by the same proportion (while, and this is important, we keep the technology unchanged).

Perhaps it could help you to revisit your example based on the above standard definitions of these concepts.


Why does the Dutch disease cause the value of the currency to increase?

How well do you know what you mean with "currency increases"?

The term Dutch disease describes an event where a single sector's boom (your example, finding oil) causes the said sector to grow and other sectors to decline as revenues increase in one sector. The booming sector causes the value of the currency (for example, the Dutch guilder) to appreciate (grow in value) compared to other currencies. This makes exports from non-booming sectors more expensive for other countries and therefore the booming of one sector causes a negative competitive effect on the other sectors.

So the effect is:

Oil is found
Oil (manufacturing) sector booms
Currency value of the Dutch guilder increases as demand for Dutch guilder increases (to pay for Dutch oil. Can cause/be due to an inflow of foreign currencies)
Exports from other sectors become more expensive (due to currency value increase)
Other sectors become less competitive.


Questions about the gold standard and exchange rates

I would interpret the “Gold Standard” as being all (major) countries each pegging their currency to gold, or at least to a currency pegged to gold (e.g., the U.S. dollar in the Bretton Woods system). A single country pegging its currency to gold would make gold investors happy, but it would do very little otherwise, since the currency would just float like gold prices already do.

This means that the exchange rates cannot appreciably move (there is usually a small trading band around parities).

This is why that DeLong says that the decline “... would not be allowed.” That is, if tge currency was weakening within its small trading band, the central bank needs to defend the parity: typically raising interest rates. If that does not help, the government is forced to launch fiscal austerity, which shrinks the economy. This reduces imports, which will hopefully take the pressure off the exchange rate.


How do we find pareto optimal points in a 2 goods simple exhange economy?

The following plot has the answer to your question, observe it carefully:


Additive multiattribute nonlinear utility functions

These are sometimes known as "quasi-separable utility functions".  A recent paper that describes the theory is Quasi-Separable Utility by Qin and Rommeswinkel.

Utility of this form arises fairly naturally in macro and finance.  If your current utility depends on lagged consumption as well as current, then you get utility of this form,
$$
\sum_{t=1}^\infty u(C_{t-1}, C_t),
$$
where $C_t$ is time $t$ consumption.  One particularly common form is
$$
\sum_{t=1}^\infty u(C_{t} - h C_{t-1}),
$$
which is known as habit utility, since it captures (for $h > 0$) the idea that get used to a certain level of consumption, so if it drops you are very unhappy.


Interpretation of coefficients in a regression with a lagged dependent variable

tldr If $\mathbb{E}[\varepsilon_{i,t}|X_{i,t-1}, Y_{i,t-1}] = 0$ then the coefficient $\beta_2$ is equal to:
$$
\frac{\partial \mathbb{E}[\ln Y_{i,t}|X_{i,t-1}= x_{i,t-1}, Y_{i,t-1} = y_{i,t-1}]}{\partial \ln x_{i,t-1}}
$$
If $\varepsilon_{i,t}$ is independent of $X_{i,t-1}$ and $Y_{i,t-1}$ (which is a stronger condition) then it is also equal to:
$$
\frac{\partial \ln \mathbb{E}[Y_{i,t}|X_{i,t-1}= x_{i,t-1}, Y_{i,t-1} = y_{i,t-1}]}{\partial \ln x_{i,t-1}}
$$
In both cases, they give an estimate for the elasticity of employment with respect to productivity yesterday (conditional on employment yesterday), i.e. what is the (expected) percentage point change in employment (today) due to a 1 percentage increase in productivity (yesterday) given the level of employment yesterday.
estimating elasticities
If
$$
\ln y = \alpha + \beta \ln x,
$$
Then the $x$-elasticity of $y$ is given by:
$$
\frac{\partial \ln y}{\partial \ln x} = \beta.
$$
It measures the percentage change in $y$ due to a 1 percent increase in $x$.
Now let's go to the stochastic framework:
$$
\ln Y = \alpha + \beta \ln X + \varepsilon
$$
And assume, as usual, that $\mathbb{E}[\varepsilon] = 0$.
The elasticity of $Y$ with respect ot $X$ is not really defined as $Y$ is not a function of $X$ (i.e. both are random variables).
There are two natural generalizations of elasticity in this setting:

$\dfrac{\partial \ln \mathbb{E}[Y|X = x]}{\partial \ln x}$, which is the elasticity of the conditional mean function $\mathbb{E}[Y|X = x]$.
$\dfrac{\partial \mathbb{E}[\ln Y|X = x]}{\partial \ln x}$, which is the derivative of the conditional mean function $\mathbb{E}[\ln Y|X = x]$.

Because the log is a non-linear transformation, the two are not necessarily the same.
However, if $X$ and $\varepsilon$ are independent then they are identical and equal to $\beta$. To see this, notice that for 1.
$$
Y = e^\alpha  X^\beta e^\varepsilon
$$
Then taking conditional expecations and using the independence between $\varepsilon$ and $X$:
$$
\mathbb{E}[Y|X = x] = e^\alpha x^\beta \mathbb{E}[e^\varepsilon|X = x] = e^\alpha X^\beta \mathbb{E}[e^\varepsilon],\\
\to \ln \mathbb{E}[Y|X = x] = \alpha + \beta \ln x + \ln \mathbb{E}[e^\varepsilon].
$$
Taking the partial derivative with respect to $\ln(x)$ gives:
$$
\frac{\partial \ln \mathbb{E}[Y|X = x]}{\partial \ln x} = \beta.
$$
For 2. we immediately have:
$$
\mathbb{E}[\ln Y|X = x] = \alpha + \beta \ln x + \mathbb{E}[\varepsilon|X = x] = \alpha + \beta \ln x.
$$
So:
$$
\frac{\partial E[\ln Y|X = x]}{\partial \ln x} = \beta.
$$
If only $\mathbb{E}[\varepsilon|X] = 0$ but $X$ and $\varepsilon$ are not independent, then only the derivation of 2 is valid and 1 and 2 are not necessarily equal.
If you want to estimate the elasticity of the growth rate, then the right regression is indeed:
$$
\ln(Y_{i,t}/Y_{i,t-1}) = \beta_0 + \beta_1 \ln X_{i,t} + \varepsilon_{i,t}.
$$


You have different questions in your title and body I will try to answer them separately:

Are there western governments that don't have to borrow in their own currency?

Yes, all western governments that I know of are allowed to borrow in any currency they want. They usually borrow in their own currency because it is easier for them to repay the debt if they borrow in currency they can control.

Are there any western and/or democratic countries where the central bank buys all government debt, interest free, directly from the treasury?

I do not know of any western government that would get sell this debt interest free  de jure (whether on primary market or not). However, de facto all interest payments that are collected by central banks are send straight back to the government treasury,  so any interest payments that government pays on government bonds are de facto just free and just money juggling - like when government employees have to pay taxes (government first spends money on paying their salaries and then takes some of that money back in taxes instead of paying the post tax salary).

are there any such countries where government spending is effectively financed by money printing without the intermediary of a private market for government bonds?

There are some countries that are not autocratic and do that but are not 'western' countries either. For example, Indonesia is democratic country and their central bank recently started to purchase bonds on primary market.
However, some European central banks are considering it. For example, Bank of England (BOE) recently stated it will:

"keep under review the case for participating in the primary market”, suggesting it could eventually buy bonds from the government or companies directly rather than on secondary markets as it does now.

Nonetheless, this has nothing to do with the interest payments themselves, they will likely not be eliminated (and why would it even matter given that the government essentially pays that interest to itself?).
Fed now has a program to buy bonds on primary market from companies, but not from government.
Czech republic recently tried to fast-track a bill that would allow their central bank to do that, but only during financial crises, but the motion failed. However, it is possible the bill will pass in the future (this was only fast-track process that would skip debate with experts/scientists and public debate).


Why are constant price elasticities reported?

Premise of your question is simply false. You state (emphasis mine):

Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity.

Incorrect. Consider trivial counter example. A perfectly reasonable demand can be given by:
$$ Q  = A p^{-\epsilon} $$
where $Q$ is quantity, $p$ price and $A$ and $\epsilon$ are constant parameters.
Hence the price elasticity of demand $e$  will be given by:
$$e = \frac{dQ}{dp} \cdot \frac{p}{Q} = -\epsilon  A p^{-\epsilon-1} \frac{p}{ A p^{-\epsilon}} = -\epsilon $$
Consequently, clearly elasticity does not always depend on the levels of price and quantity and can be completely constant.

Then why is everywhere (research papers, textbooks etc.) a constant estimate provided?

Again false premise, in several textbooks and papers you will find elasticity being expressed as a function of price and quantity. Since you made statement about everywhere it is enough to show one example to the contrary which would be Varian Microeconomic Analysis, where there are several examples where elasticity in a model won't be constant.
In empirical papers elasticity will be most often estimated as a single number, but this is due to the way how empirical models are set up. It is too broad to discuss all empirical models in a single question, but typically coefficients estimated by model are values that hold on average. For example, in a regression $y_i = \beta_0 + \beta_1 E_i + e_i$ where $y$ would be income and $E$ years of education, the $\beta_1$ would tell us how much on average $y$ increases with education, it would not tell us what happen to every individual kids. That is if $\beta_1=10$ that does not mean that if your kid stays in school one more year they are guaranteed that their income will increase by 10. That is just a value we would expect for average kid.  Similarly, even if researcher assumes elasticity is not constant because they are using linear demand, they might be forced to estimate elasticity at a point rather than as a whole function and convention is to estimate point elasticity at average quantity and price (see discussion in Espey  Espey  & Shaw (1997).


OPEC's market share strategy: What is the benefit of market share in a fungible market?

The benefits of market share in the somewhat-fungible oil market:

pricing power (because OPEC is big enough to retain pricing power in the long term because they're a big % of the market. They're just choosing to not use that pricing power right now, perhaps because they're choosing to bankrupt competitors)
geopolitical strength

Note that there are additional explanations:

Saudi is one of the lowest short-run-marginal-cost producers, so you'd expect them to be one of the last to cut production, just through normal business, regardless of the other stuff
we know that most known fossil-fuel reserves are going to stay in the ground, so reserve-owners are now under pressure to try to ensure that theirs aren't, and someone else's are.
You could view the last few years as a sucker-trap: high oil prices lured in lots of investment into new production outside OPEC. An oil price below $30 bankrupts many of those new entrants, who have high debt-servicing costs and high production costs. And because of that, future investors will be much more wary, and require much higher returns
Saudi knew that the international sanctions on Iran were on the way to being lifted. If Saudi had cut oil production, it would have weakened its market strength (and thus also its political strength) relative to Iran upon its return to the international oil market.


Income effect for inferior goods question

Could you clarify what your your question is? If Ramen is inferior then an increase in wealth (parallel shift of the budget line) should cause the consumption of ramen to go down. Hence $B$ should be to the left of $A'$. Then income effect is the distance on the x-axis from $A'$ to $B$.

(Edit in response to the comment)

Let me try.

The slope of the budget line is related to ratio of prices. So $A'$ is the point of tangency of between the old indifference function and a budget line with a new ratio of prices. Essentially what it is saying is what would be your basket if the price of ramen was reduced but some of your money was taken away, so that you are no better and no worse off. The change from A to A' is pure substitution effect.

Then, since your money was not actually taken away from you, you have to consider income effect. This is achieved by shifting the dased line upwards. So A -> A' is substitution effect. A' -> B is income effect.

Direction of substitution effect is always the same. Higher relative price means less consumption. Direction of income effect depends on the type of goods.


Comparative data on agricultural subsidies

Which countries are you looking at? This paper seems to suggest there's data from OECD countries.

You might also take a look here and here. I downloaded zip files and, if I understand correctly, there seem to be data of subsidies by country, product, and year.


Can inflation occur in a positive-sum game currency system such as the Stack Exchange reputation system?

In the Stack Exchange reputation "economy", we can think of the reputation points as "money". And the "goods" are:

Downvotes — cost 1 point each.
Various privileges — zero cost, but unlocked only when the user reaches various thresholds.

To my knowledge, these prices (1-point cost of each downvote and privilege thresholds) have never changed. And so, there has never been inflation. What has merely happened is that "wealth" (the total stock of reputation points) has kept increasing.

Response to Pere, who writes:

  Newer sites have lower thresholds for several privileges - for practical reasons - and those thresholds rise as the site expands. For example, being able to create tags "costs" 150 reputation in Economics SE, 300 reputation in Cross Validated SE and 1500 reputation in StackOverflow. Therefore, privilege prices are subject to inflation as sites grow.

Each individual SE site is its own individual "economy" that is sealed off from all other SE sites.¹

And so, whether "prices" differ across SE sites does not matter. What matters is whether "prices" have changed over time, within each SE site.

To my knowledge, the privilege threshold "prices" on Economics SE have never changed.

And it also seems that for the original, flagship site (Stack Overflow), "prices" have also not changed for many years. We can, for example, compare today's "prices" with those from 2013:

As you can see, all prices have remain unchanged since 2013. There has therefore been zero inflation since (at least) 2013. The only change is the introduction of a "new good", namely "Access to site analytics", which is priced at 25,000 points.

Response to @kbelder, who writes:

  What about bounties? I bet those have historically trended upwards in value

The "prices" of bounties have likewise been fixed. Bounties may be offered at "prices" of 50, 100, 150, ..., and up to 500 points. I believe these prices have been fixed since the inception of Economics SE and at least since 2013 for all SE sites.

Moreover, bounties are pure transfers from one user to another.² It may be true that thanks to increased overall "wealth", the average value of a bounty has risen (i.e. in the past, most bounties were offered at close to the minimum price of 50; whereas today higher prices are offered). However, we do not consider this "inflation". Instead, we consider this to be real wealth effects.

Analogy: Say that in 1919, the New Yorker on the street might on average give \$0.20 (in 2019 dollars) to a homeless person. In 2019, this might instead be \$1 (in 2019 dollars). The average real transfer from a New Yorker to a homeless person has risen five-fold in real terms. This is not inflation. Instead, it is a consequence of increased overall wealth.

Footnotes.

To my knowledge, there is one thing that links reputation points across the various SE sites. This is the automatic +100 points for established user accounts that are "trusted" on the SE network. I'm not sure how this works. But I would say that these +100 points are relatively inconsequential and may be safely ignored. They would perhaps be the real-world equivalent of an immigrations officer giving a "trusted" visitor \$1 the first time that visitor arrived in that country.
There are actually three things that can happen to a bounty. First, the full bounty is awarded (I believe this is the most common case). Second, half the bounty is awarded to the highest score (provided that answer has a score of at least +2 — see Economics.SE bounty rules here). Third, no bounty is awarded. In the latter two cases, we do not quite have a pure transfer; instead, we have a leaky bucket where either half or all of the points are "lost to society".


How do I determine the demand curve of a product through a survey?

If it is a small project, you may benefit from reading this

At most, you can estimate price elasticity of demand using your survey data to make predictions/forecasts. For this you need to estimate a demand function as @EconJohn suggested, but you need to include additional variables such as income, price of a substitute or complement etc. to hold them constant (i.e. ceteris paribus). Note that the demand function does not allow you to draw the downward sloping demand curve (like the one in textbooks). It is very hard, in practice, to draw actual demand curves using survey data. Even when you have the most complete data, without including the suply function in your estimation, you face so called identification problem. In short, econometric estimation of demand curve does not allow you to draw a downward sloping curve of the type you see in textbooks.

The link above has a good explanation about this point, and it can be used as a guide for your project.


Doubt on a simple problem from Mankiw's Macroeconomics

In theory, yes. If there are perfect** (and mutually exclusive) substitutes, and exactly the same amount of money is spent on those substitutes (and individuals would not have spent that money otherwise), then GDP could remain constant.However, even for extremely simple goods, this is unlikely.

However, the question asks how the events are likely to affect GDP. In practice, there are relatively few true perfect substitutes. If people can't go to Disney world, some may opt for less expensive alternatives, like watching a movie. Another consideration is whether or not Disney is obligated to pay staff during the time (probably not, so now there are people who are making less money and buying fewer things), Disney isn't purchasing food stocks and cute teddy bears for its canteens and games, and a host of other effects. The point is, it is most likely to push GDP down.

In this case, I would say the question kind of implies "all else held constant". The key learning outcome here is "When there is a shock, and firms are forced to cease production temporarily, this negatively impacts GDP." This is the basic principal for first year econ. In later years, you will get into general equilibrium, which deals more with the side-effects of such shocks and aggregate micro-decisions.

** Edit: It is technically not strictly necessary for them to be perfect substitutes (see comments below). But in practice (and in the spirit of the first-year level question), shutting down a whole firm/industry will push GDP down. People are choosing between consumption and savings when deciding how to spend their money, between labor and leisure when deciding how to make their money, and taking away a large source of consumption-based utility is likely to ultimately shift choices away from consumption and towards savings and leisure both of which push down GDP in the current period, though savings generally increases it for t+1 (I am not going to get into trade-deficit effects here). But that is a more complex analysis that isn't really the point of the thought exercise in this particular case. Key learning: External shocks that reduce production generally reduce GDP, all else held constant.


How to calculate the optimal wage and the optimal rent on capital in an economic model?

I hope this answers your question (if not please let me know!), but the reason they get their wage and rental rates:

$$  w_t = \frac{\partial y_t}{\partial l_t} \text{ and } r_t = \frac{\partial y_t}{\partial k_t} $$

(I assume) is for the same reason most papers use that equation. Consider the firm's problem, that they want to maximize profit, which is (total) revenue ($p_t \times y_t$) minus costs (which we can consider as the total cost of labor, which is $w_t \times l_t$, and capital, which is the rental cost of capital times capital used, $r_t \times k_t$). Therefore, we're trying to maximize:

$$\max_{l_t,k_t, w_t, r_t} \{ p_t y_t - ( w_t l_t + r_t k_t ) \}$$

If we normalize price of the output to $1$, we can solve for the optimal prices of labor and capital by solving the above problem. Taking the inputs one at a time, we get the first order conditions for labor:

$$0 = \frac{\partial y_t}{\partial l_t} - w_t$$

Which, of course, rearranges to

$$w_t = \frac{\partial y_t}{\partial l_t} $$

and for capital:

$$0 = \frac{\partial y_t}{\partial k_t} - r_t$$

which rearranges to:

$$ r_t = \frac{\partial y_t}{\partial k_t} $$

Of course, these are pretty standard derrivations across different fields of economics, so their exact reasoning might be slightly different, but this methodology should provide some of the clearest insight into why that's the case.


Let us assume all these functions exist and are differentiable. Furthermore let us denote the inverse of the function $\overline u^1(x_1)$ by $x_1\left(\overline u^1\right)$.

Note that for any invertible function $f$ we have
$$
\left.\frac{\text{d} f(x)}{\text{d} x}\right|_{x=x_0} =
\frac{1}{\left.\frac{\text{d} f^{-1}(y)}{\text{d} y}\right|_{y=f(x_0)}}.
$$
In our case this means (with slightly simplified notation)
$$
\frac{\text{d} x_1(\overline u^1)}{\text{d} \overline u^1} =
\frac{1}{\frac{\text{d} \overline u^1(x_1)}{\text{d} x_1}}.
$$
We will further simplify notation by writing
$$
\overline u{^i}' =  \frac{\text{d} \overline u^i(x_1)}{\text{d} x_1}.
$$
We will denote second derivatives similarly by affixing $''$.

The question is under what conditions
$$
\overline u^2\left(x_1\left(\overline u^1\right)\right)
$$
is concave in $\overline u^1$. This of course, depends on the sign of
$$
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2}.
$$
Differentiating
\begin{align*}
\frac{\text{d} \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}^1} & =
\frac{\text{d} x_1(\overline u^1)}{\text{d} \overline u^1}
\frac{\text{d} \overline u^2(x_1)}{\text{d} x_1}
=
\frac{1}{\overline u{^1}' }\overline u{^2}'
=
\frac{\overline u{^2}'}{\overline u{^1}' }.
\end{align*}
Going one step further
\begin{align*}
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2} & =
\frac{
\frac{1}{\overline u{^1}' }\overline u{^2}^{''} \overline u{^1}'
-
\overline u{^2}'\frac{1}{\overline u{^1}' }\overline u{^1}^{''}
}
{\left(\overline u{^1}'\right)^2}.
\end{align*}
This implies that the Pareto-frontier is concave when
\begin{align*}
\frac{\text{d}^2 \overline u^2\left(x_1\left(\overline u^1\right)\right)}{\text{d} \overline {u}{^1}^2} & \leq 0 \\
\\
\frac{
\frac{1}{\overline u{^1}' }\overline u{^2}^{''} \overline u{^1}'
-
\overline u{^2}'\frac{1}{\overline u{^1}' }\overline u{^1}^{''}
}
{\left(\overline u{^1}'\right)^2} & \leq 0 \\
\\
\overline u{^2}^{''}
 & \leq \frac{\overline u{^2}'}{\overline u{^1}' }\overline u{^1}^{''}
\end{align*}
for all $x_1$.


Is comparative advantage undermined by a welfare state?

The basic result about the effect of trade on income distribution is Stolper-Samuelson in the Heckscher-Ohlin model, which says that trade liberalization is associated with a rise in the prices of the factors of production that are used intensively in the production of goods the country has a comparative advantage in, and vice versa. If you take "skilled" and "unskilled" labor to be two different factors of production, for instance, Stolper-Samuelson says that a country with abundant skilled labor lifting trade barriers should lead to an increase in income inequality. A welfare state committed to keeping relative incomes of workers within a certain range may need to raise distortionary taxes to expand various income transfer programs to "unskilled" workers in this case, which could lead to inefficiency for various reasons. (Even if the taxes used to finance the income transfer are lump-sum, as long as the benefits phase out this leads to high effective marginal tax rates on the lower end of the income scale, i.e a disincentive to work and to acquire new skills.) In this sense, it's indeed true that a generous welfare state can reduce the efficiency gains from free trade.

The rest of the answer is a rant about the concept of "strategic market dumping" that was brought up in the comments, so if you're not interested you can skip that part.

Things like "strategic market dumping" are ill-defined concepts that are usually used to justify tariff increases without having much backing in theory. If "dumping" is expected to go on for a finite period of time and the associated good can be stored at no cost, for instance, then in a perfect foresight world there's an arbitrage bound on the price of the good today which forces it to grow at a rate less than or equal to the real interest rate available in the broader economy. Adding storage costs does not change the basic result very much. In that case, "dumping" is not a good strategy to begin with - if you try to do this, people will buy up the goods you're selling at low prices, hold them until you reverse your policy, and then flood the market with goods they have stored up in their inventories once you stop the "dumping", leaving you worse off. To justify a policy of industry protection, you have to argue that the government can see the reversal of the dumping policy coming and private agents can't - in this case, the real return on an investment in producing and storing oil would be higher than the market real interest rate is saying it will be, so the market outcome will be suboptimal. It's easy to justify any government intervention by saying that the government is clairvoyant and private agents are not, however.

What about labor market (or, more generally, factor market) frictions? Isn't the point of "dumping" to eliminate your competitors through "artificially low" prices which force them to go out of business? Well, here you're making the mistake of assuming that the government knows about labor market frictions and the private sector doesn't, therefore the government should do something about the situation. In real life, the private sector is well aware of the search frictions in the labor market, the difficulty of firing workers in some countries, etc. This means that if they expect the "dumping" strategy to be reversed in the future, they will choose to borrow money or raise capital to stay in business. They will downsize in response to the "dumping" policy, but not as much as you think they would, and certainly not as much to make a government "protection" of the industry a good policy.

Does this mean markets can never allocate resources poorly? Of course not - if the country doing the dumping manages to fool private agents into thinking their "dumping" policy was permanent while it was temporary, then the market allocation will be suboptimal. However, in hindsight every price and every allocation is wrong. We certainly don't expect market allocations to be optimal ex post, so it's not surprising to see cases where it seems like the market allocation was poor and led to many workers being stuck in jobs where wages fell due to international price changes of goods. In fact, this would happen in a stochastic world where markets were functioning perfectly and giving you an ex ante Pareto optimal allocation of resources. Even a social planner with access to all information available at a given date and the mental capacity to solve the associated economic calculation problem would act in ways that sometimes seem foolish ex post. This doesn't mean the social planner is not doing his job properly, it just means there are things about the future that even a social planner can't know with perfection. The argument is no different for market allocations - allocations that are poor ex post do not justify government intervention ex ante.


Profit Maximisation if MC is still falling after intersecting MR

Since the OP "recovers old-ground", let's turn the flashlight on some dark spots of this ground.

It is almost "automatic" to think that "marginal revenue" is the price at which the additional unit of output will be sold 9so that it is "the revenue that the additional unit brings in"), and that "marginal cost" is what this additional unit will cost. We should remember that this only holds when the firm is a price-taker, and its choices on output level does not affect the price of output, nor the prices to be paid for input factors. For if

$$TR(q) = pq \implies MR = p$$

but if

$$TR(q) = p(q)\cdot q \implies MR = p'(q)q + p < p$$

since $p'(q) <0$

The same goes for Total Cost, where denoting by $\mathbb w$ the prices commanded by input factors

$$TC(q) = C(\mathbb w, q) \implies MC = \frac {\partial C}{\partial q}$$

but if

$$TC(q) = C(\mathbb w(q), q) \implies MC = \frac {\partial C}{\partial \mathbb w}\cdot \frac {\partial \mathbb w}{\partial q}  + \frac {\partial C}{\partial q}$$
(I have avoided matrix-differentiation notation). Note that the effect of increased quantity on input prices is not clear-cut: in the usual monopsonistic setting, it will tend to decrease the unit cost of the inputs -but if the currently supplied quantity of input has reached a ceiling, it may push prices up (although that is considered more of a short-term effect).

The usual setting, and the one indicated by the OP, is to assume price-taking behavior in the markets for inputs (in which case the non-constancy of marginal cost stems purely from technological factors), while assuming some monopolistic power in the product market, i.e. a downward sloping demand curve.

So it is more general to think "marginal revenue is the increase in total revenue if we increase quantity by one unit", and "marginal cost is the increase in cost if we increase quantity by one unit", disassociating the concepts from the revenues and the costs of the "additional unit" itself.

Having cleared this, the OP asks

  But suppose MC has a minimum beyond the point at which MR=0. So MR
  still intersects MC from above but MC is still falling (does this
  still satisfy SOC?)

This is something like

which looks pretty standard. As to whether Second-order conditions hold, in this post one can find a mnemonic about slopes, since the SOC is phrased in terms of derivatives of the curves shown in the above graph.

  In that case would the max difference between TR and TC be greatest at
  some Q higher than where MC=MR?

Since at $MR=MC$ the marginal revenue curve is assumed to cross the marginal cost curve from above, it follows that for higher quantities, marginal cost will be higher than marginal revenue. Given the previous suggestion as to how it is best to think about Marginal Revenue and Marginal Cost, I believe the answer to the question is easy.


Any two specific goods can be perfect substitutes, imperfect substitutes or even complements depending on the preferences. It's not goods themselves, but our preferences towards them what make them substitutes or complements.

However, when many people have similar preferences towards those goods, it is common to characterize goods as being, for instance imperfect substitutes. Apparently, you have no qualms about Pepsi and Coca-Cola being imperfect substitutes, and I guess that's because it agrees with your own preferences towards them. Tea and Coffee are imperfect substitutes for many people, who may drink sometimes one and sometimes the other. Of course, they're not substitutes at all for people who only drink Tea or only drink Coffee. The fact that we colloquially say that they are imperfect substitutes should be understood as most people thinking of them as such.


Interest payments are contractual payments. They need to be specified with an exact formula. For a given set of cash flows, one could come up with a multitude of formulae that end up giving the same values. In order to make interest rates comparable, laws, regulations, and market conventions specify that different conventions be used.
Of these conventions, simple and compound interest are possible choices. For simple interest, the interest payment is just the interest rate times the principle times a time factor. Compound interest takes into account increases in principal.
If a loan has associated interest payments that must be paid immediately, the loan resembles a generic bond: principal amount does not change. Bonds have a wide variety of conventions to convert the price of a bond to a yield, including a “simple interest rate” convention. However, changing the convention does not change the contractually defined price or cash flows. Since generic bonds do not roll interest payments into the principal, “compound interest” does not make sense.
For loans, interest is often rolled into the principal. We can then use simple interest or compound interest conventions.
Assume a monthly frequency, with no calendar date irregularities, for a loan agreement.

The interest rate could be specified as a simple interest rate of 1% a month.
The interest rate could be specified as an annual compound interest of $1.01^{12}-1$ or 12.68% (approx).

Regardless of the convention used, if the borrower makes no payments for a year, the principal will grow by (roughly) 12.68% since the lender will roll the interest into the balance at the end of the month, and then make the simple interest calculation based on the new balance. That is, lenders will still charge interest on interest in practice.
Which one to use is purely a cosmetic decision - is it easier to work with 1% a month, or 12.68% a year?


In general, stock picking does not work.

For more information, see this keynote address by Daniel Kahneman, who found that stock-pickers perform worse than chance when they change stocks: https://forum.amundi.com/the-psychology-of-investors-the-cost-of-ideas/

Here is another reference on the subject:

  Abstract

  Is expertise portable across closely related domains? We investigate this question empirically using a unique dataset of market
  experts---institutional investors with portfolios averaging $573
  million. A striking finding emerges: while there is clear evidence of
  skill in buying, the investors' selling decisions underperform
  substantially--even relative to random selling strategies. This holds
  despite the similarity between the two decisions in frequency,
  substance and consequences for portfolio performance. We present
  evidence that an asymmetric allocation of cognitive resources such as
  attention can explain the discrepancy: investors are prone to a
  systematic, costly heuristic process when selling but not when
  buying.

  Akepanidtaworn, Klakow and Di Mascio, Rick and Imas, Alex and Schmidt, Lawrence, Selling Fast and Buying Slow: Heuristics and Trading Performance of Institutional Investors (May 8, 2019). http://dx.doi.org/10.2139/ssrn.3301277


As @FooBar mentions in a comment, with "Income" we want to measure what we produce (production being the whole targetfocus of economic activity, with consumption being the ultimate purpose).

Measuring Income as

  "If you think of income as the inflow in your bank account."

may appear an adequate approach to measure it for some situations: if I work and my employer pays me by bank-transfers, all my income is the sum of these inflows (I am sure that the OP meant to exclude other kinds of inflows, like for example from one of my accounts to another).

So, If I sell something that belongs to me, doesn't that also increase my Income, and so National Income?
No, it restructures your wealth in terms of liquidity: your non-liquid wealth decreased while your liquid wealth increased. You did not produce anything, you just liquidated part of your wealth.

Buying a football, is buying a durable consumer good. You don't get to eat the football, or burn it at a temple (at least in normal usage), you make use of its bouncing services to entertain yourself. So while you hold on to the football and you use it, it is a utility-enhancing capital good, and so part of your wealth.

The very fact that you were able to re-sell it, reveals that this durable/utility enhancing capital good was not fully depreciated, it still had some potential flow of services in it -this is why the other person paid you for it.

So it is selling of a capital good after this good has become wealth, and so it does not reflect production.

Mind you  that the "Used goods Sector" does create Income: it offers services around the used goods: the good "Used football in my closet" is not the same good as "Used football in a shelf for sale". But the Income they generate is the difference between how much they paid me for the football, and how much they will sell it... which naturally leads us to the concept of "Value Added" which is the one we should make certain to understand accurately, in order to navigate in quantitative economics.


Demand functions in merger simulation models

In your case, you might try to reach the said antitrust authority and ask for the dataset. You also could reach the parties themselves and ask for the shared data. Authorities acquire these datasets through parties BI data and/or third parties that might have some connection to the merger (companies concerned with the operation, clients etc). I believe it is highly improbable that the data is made publicly available anywhere.


Calculating the Competitive Equilibrium in a pure exchange economy with 3 comodities and 2 agents

First, there is no such thing as the competitive equilibrium price vector. If $(p_1,p_2,p_3)$ is a competitive equilibrium price vector, so is every positive multiple of this vector.
Second, to find the demand you also have to look at the cases where some inequalities are not strict, where they are actually equalities.
Here is how you can simplify the problem: The equilibrium consumption must be at least as good as keeping the endowment for both consumers. You can check that this means that if one of the consumers consumes the total endowment of a single good and nothing else, they are worse off. So in equilibrium, everyone must consume a positive quantity of at least two commodities. It is also easy to check that for no price system would both consumers consume positive amounts of all three commodities. There is no relative price $p_1/p_3$ under which both consumers consume positive amounts of both good $1$ and $3$. Togehter, this should help you pin down the (here unique) relative equilibrium prices.


Generally speaking, shouldn't GDP roughly increase equivalent to population growth?

Basically the answer is yes

Population growth is a key factor in GDP growth. Population creates both demands for goods and labour supply

Unlike GDP, GDP per Capita, cannot, obviously rely on population growth, and that is (or at least should be) the goal of governments. and unlike population, the "rest of the factors" affect GDP per Capita just the same

A couple of extra points besides the obvious: population growth, by births or by immigration, affect age distribution, which affects both supplies of labour amd demands for goods and services

Because of it, population stagnation may mean decrease in GDP per Capita and long term economic issues


Reasons for the recent rise in long-term unemployment

I would "advise" to wait... a few more years! From the figure you provided, there is a perhaps slow but visible rising trend (with business-cycles-like "seasonality") for both the long-term and the mid-term unemployment rate, a fairly stable trend in terms of steepness over 60 years. The sudden peak of long term unemployment in 2009 is impressive but it also fell sharply in the next three years. To me, it appears critical to know the unemployment rates for the next four-five years, in order to decide whether we have a structural break with the historical rising trend becoming more steep, or just a few-years-incident due to the crisis.


Why a non-autonomous increase in aggregate demand will not trigger muliplier effect?

Because if the increase doesn't come from an autonomous component, it already amounts to the non-autonomous demand increase + multiplier.


Doesnt the fact that cash reserves go down in recessions contradict Keynesianism?

You very often see Krugman an other economists imply that corporations holding cash is a bad thing or affects demand.

http://krugman.blogs.nytimes.com/2011/07/02/cash-is-not-the-problem/?_r=0

However, you have to remember that Krugman is just stating his personal opinion. He does sometimes say things that contradict academic Keynesianism.

The fact that corporations hoard less cash during recessions doesn't necessarily contradict Keynesianism or New Keynesianism, but it does require these theories to explain why corporations behave this way (which the theorists have not attempted).