Inflation, Future, And Value of Money - deciding to buy a house I live in Asia where the inflation rate is approximately around 4.5-6% (average) in last 10 years. If in the next 10 years I plan to buy a house in other country (any country which I have the currency). Should I choose to buy the currency which have the lowest inflation rate?

For example, if I hold US Dollar 100,000 right now, and I can buy a house. But with the inflation rate of 1.3% in 10 years the price will be approximately 13% (USD13,000). Means my money will not enough and I must add 13,000.

However, If I hold JPY 11,000,000 right now and I can buy house too right now, with the inflation rate of 0.1%. it will be only 1% increase in next 10 years (about 110,000 JPY). It's not so much.

So, Is my idea correct?

If a country has a lower inflation rate than your country, it's currency will tend to appreciate against your currency over time, see interest rate parity.

It is true that if inflation is low, house prices will tend to increase less over the next ten years, however, the currency will tend to appreciate, making it more expensive to buy a house in ten years time.


How to find $\phi$, that denotes the correlation of signals among informed traders? Since I do not have an answer on Quantitative Finance in my question I cross-post here the problem to tag some other categories
The following assumptions are part of the paper of Back, Chao and Willard and I can not solve for the statistic that is denoted as $\phi$ in the sequel. I would be glad if anyone could help me. Below i set the assumptions and the equations of interest
Suppose that in the market, there are $N\geq 1$ informed agents, who trade a risky asset continuously in the time interval $[0,1)$. Each agent $i$ receives a mean-zero signal $\tilde{s}^i$ at time 0. We assume the signals and the liquidation value of the asset have a nondegenerate joint normal distribution that is symmetric in the signals. Symmetry means that the joint distribution of the asset value and the signals $\tilde{s}^1,...,\tilde{s}^N$ is invariant to a permutation of the indices $1,...,N$. Let $\tilde{v}$ denote the expectation of the liquidation value conditional on the combined information of the informed traders. By normality, $\tilde{v}$ is an affine function of the $\tilde{s}^i$. By rescaling the $\tilde{s}^i$  if necessary, we can assume without loss of generality that
\begin{equation}\tilde{v}=\bar{v}+\Sigma^{N}_{i=1} s^i\end{equation}
for a constant $\bar{v}$. For simplicity, we assume $\bar{v}=0$. Let
\begin{align}\phi=\frac{var(\tilde{v})}{var(N\tilde{s}^i)}\end{align}
The statistic $\phi$ is a measure of the quality of each agent’s information. Specifically, it is the $R^2$ in the linear regression of $\tilde{v}$ on $\tilde{s}^i$, that is, it is the percentage of the variance in $\tilde{v}$ that is explained by the trader’s information.
Letting $\rho$ denote the correlation coefficient of $\tilde{s}^i$ with $\tilde{s}^j$ for $i\neq j$, one can compute $\phi$ for $N>1$ as
\begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}
If $\phi=1$, then either $N=1$ or the $\tilde{s}^i$ are perfectly corellated. In either case each informed trader has perfect information about $\tilde{v}$.
My questions are the following

what does it mean intuitively "a nondegenerate joint normal distribution" and in particular I would like to understand the term nondegenerate.
What does it mean "invariant to indices" ?
the liquidation value is equal to the sume of the signals, does this come from the assumption that it it an affine function of the $\tilde{s}^i$?
How do we find that measure $\phi$? is it from the linear regression of $\tilde{v}$ on $\tilde{s}^i$?
How $\phi$ is tranformed to
\begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}

Here it is a link from the paper

A degenerate joint normal is distribution is one in which you cannot find a PDF for the distribution. They assume you can. (The covariance matrix is invertible).

Let $f(s_1,s_2\dots,s_n,v)$ be the distribution.
If I was to exchange $s_1$ for $s_2$, $f(s_2,s_1,\dots,s_n,v) = f(s_1,s_2\dots,s_n,v)$, the distribution does not change. And you can exchange as many indices (signals) as you'd like.

The assumption that the distribution is jointly normal implies $\tilde{v}$ is an affine function of $\tilde{s}_i$, which then implies the sum of signals.

Regress $\tilde{v}$ on $\tilde{s}_i$ and calculate $R^2$ (the Coefficient of determination).

This is something I hope someone else can answer.


Why is the Stock Market Cap / GDP ratio on FRED well out of date? FRED posts the Buffet Ratio here:
https://fred.stlouisfed.org/series/DDDM01USA156NWDB
But it ends in 2017, anyone know why?

This indicator is taken from the World Bank's Global Financial Development dataset, which was "last updated in September 2019 and contains data through 2017".


How does exchange rate change with respect to interests? If the central bank raises intereat rates, isn't the money supply now less and the exchange rate is greater? I have run into opposite statements and dont know how to approach this problem.
Thanks in advance for everyone's help

If by “greater”, you mean higher number It depends on how you quote exchange rate. Under American system the exchange rate between USD and euro would be given by $S=EUR/USD$ under European system the exchange rate would be quoted other way around as $S=USD/EUR$. So under American system if the USD appreciates due to higher interest rates the $S$ would actually get lower while under the European system appreciation would actually lead to greater/higher exchange rate $S$.

Otherwise, ceteris paribus, higher interest rate has to lead to appreciation under the standard exchange rate model since the exchange rate is given in standard monetary model of exchange rates:

$$S=ln(p)-ln(p_f)= ln(m)-ln(m_f) -(ln(y)-ln(y_f))+\lambda(i-i_f) $$ (where $p$ is price level, $m$ money supply, $y$, real GDP i interest rate and $f$ denotes foreign sector - also I assume that $ ln(L(i)) \approx \lambda i$so the exchange rate should appreciate when interest rates $i$ increases and nothing else change.


Are financial markets "unique" for each "currency pair", or are they simply "translated"? This is something I've been long wondering about.
Let's say that I have statistics showing the daily closing Bitcoin price in USD since early 2009. Great. Now, if I wanted to get the daily closing Bitcoin price in SEK or EUR or whatever currency instead of USD, I could use historical exchange rate data to translate all the data points to SEK prices, or EUR prices...
Or could I?
This is based on the assumption that all the non-USD markets are simply "translated" from the USD price; that they are all interconnected. For example, if I buy X Bitcoin at some date using SEK or EUR, that is reflected in the USD Bitcoin price for that day, only translated into USD?
Or is each market (currency pair) "unique"? Maybe there are very, very few people trading Bitcoin in SEK, so the price could be completely different in SEK for one Bitcoin at any given date, differing completely from the USD Bitcoin price for that date?
I hope I've asked my question clearly. I'm trying to figure out if the data is simply "translated", or if there are unique markets for each currency which are their own "closed ecosystems".

It's not clear what level of answer you're looking for, so here is a much more basic answer.
There are indeed many exchanges with many different prices. However, if you have noticed that you could make money by exchanging your BTC for USD, exchanging the USD for SEK, and then exchanging the SEK for slightly more BTC than you started with - then someone else probably has noticed that too, and they'll make free money with the manoeuvre just as you will.
As people do this, the prices gradually equalise: you're making BTC more scarce relative to SEK so are driving its price up in SEK (making the last hop less profitable), and you're making BTC more common relative to USD so are driving its price down in USD (making the first hop less profitable).
So it is theoretically possible for such an opportunity for arbitrage to exist. In practice, however, in such a simple case, there are powerful forces acting constantly to equalise the price. As long as it's easy and low-fee to make all three transactions in the chain above (that is, as long as the respective markets are sufficiently liquid), people will make those transactions and the opportunity will naturally disappear.
If you identified a really unobvious opportunity for arbitrage (so unobvious that nobody else saw it before you) and the markets were liquid enough, then you could make a lot of money off it by exactly this manoeuvre!
In fact, very early on, the Bitcoin markets were absolutely full of arbitrage opportunities. The big players were not keen to trade in such risky places (it was routine for an exchange to just close down and run away with large amounts of other people's bitcoin - see the middle point on Forbes's article about this), so there was not such a strong force equalising the prices. As it's become more mainstream, though, more people started taking the free money, and the opportunities have dried up somewhat. So you may very well find that earlier in your dataset, the markets are much more obviously distinct.


What is causing the difference between USA financial indexes and perceived economy? While the world is facing economic problems since the spread of Covid-19 in Feb 2020, the main indexes in the world (S&P500 and NASDAQ100) reached and exceeded twice their maximum levels. I understand that:

Tech/Online companies have seen their revenue increase as a consequence of lockdowns
The pandemic is temporary and offer/demand will restore "as soon as" the pandemic is over
The Fed has purchased securities to protect the markets from excessive sell out

However, why is there such a big difference between financial markets and daily people life for the past 13 months?
I'm only 25 years old and I partially remember the MBS crisis in 2008, but I feel that market prices used to track the expansion/contraction of the economy. I have read articles claiming that economy and finance are two different things, still I am not convinced that there exists no correlation between the two.
If the Fed is protecting prices, is that a bubble waiting to explode? If the Fed is slow enough to sell to prevent a bubble, have they found a way to avoid any financial crisis in the future?

Stock market prices depend on investors view on what the value of company is which in turn depends on what is its expected future profitability. For example, under a simple Gordon growth model the stocks would be valued using (See Miskin & Eakins Financial Markets and Instiutions pp 347):
$$ P_0 = \frac{D_1}{k-g}$$
where $D$ is the dividend, $k$ required rate of return and $g$ expected growth of dividend (and dividends directly depend on profitability). The model above is a simple model and there are more nuanced ones but the point from most models is that stock prices ultimately depend on firm's expected profitability.
Typically recession lower firm profitability but rarely shocks affect all firms the same way. For example, take Covid-19. Research shows that covid-19 primarily negatively affected industries like hospitality, transport, energy and so on (e.g. see Alonso et al (2020) or Shen (2020)). On the other hand many IT/technology firms benefited from the pandemic and lockdowns. More importantly, research shows that small and medium size businesses were affected mainly small and medium size businesses (e.g. see Dua et al (2020); Dua et al (2020);  Alekseev et al (2020))*
The NASDAQ100 and S&P500 are tracking equities of 100 largest and 500 largest companies listed on the stock exchange. Consequently, shocks like covid-19 that hurt profitability of predominantly small and medium size companies that are often not even listed on stock exchanges (not even mentioning getting all the way to status 100 or 500 biggest firms respectively) will not affect stock prices as much as shocks that directly large companies, such as let's say .com bubble which mainly affected large technological businesses and correspondingly the S&P500 fell much more than during covid-19 crisis. This does not mean that covid-19 crisis was less worse for the economy than .com crisis since most firms in almost every country are small and medium size businesses. For example, according to the Eurostat in EU in 2015:

In 2015, enterprises employing fewer than 250 persons represented 99 % of all enterprises in the EU.

In the US according to SBE Council working with US 2016 census data showed that:

there were 5.6 million employer firms in the United States in 2016. ... Firms with fewer than 100 workers accounted for 98.2 percent. Firms with fewer than 20 workers made up 89.0 percent.

I think it goes without saying that typically (there might be exceptions) firms with less than 20 or even less than 100 workers simply wont make it to the S&P 500 or NASDAQ 100 or most of them might not even be listed.  Consequently, stock market is not really 'the economy', it is just a part of it.  Stock market is not a representative sample of firms, but sample of the large firms that decide to go public (e.g. small pop and mom shops would never get listed at stock exchanges because a) they do not need to raise that much capital; b) getting listed is expensive and publicly listed firms have higher regulatory burden so it does not make sense for most small firms financially). What even more, indexes such as S&P 500 and NASDAQ 100 look only at the largest firms from already sample of large firms (e.g. really the cream of the crop of the top crop). Hence these indexes are primarily affected by shocks to the profitability and expectations of the profitability of the listed large firms and there might be many macroeconomic shocks that can severely impact the economy as a whole but not as much largest firms or sometimes a macroeconomic shock that is negative for small and medium size business can be positive for large business (e.g. Covid-19 effect on restaurants vs technology giants).
Additionally,  some economists as Paul Krugman** argued that stock market is getting more disconnected from the real economy because of savings glut and because wealthy individuals have not that many other good alternatives to invest in since bonds offer little return due to low interest rates.
Lastly, Fed's job is not to take care of the stock market prices. Fed pursues loose monetary policy (e.g. low interest rates, QE and so on) because that is the textbook 101 advice for central bank to purse in recessions (e.g. see Mankiw Macroeconomics). That this leads to asset price inflation is more of an side effect of the 'cure' for the real economy not the primary intent of Fed.

 * Although an important caveat is that most of this research is preliminary as Covid-19 is recent event.
 ** However, this is opinion piece, consequently even if it is made by Nobel Prize winning economist please take it with extra grain of salt.


Why is the nature of graph of utility function different from indifference curve? I am new to Economics, but I have this doubt. The indifference curve and utility function both have the same equation, so their graph must also be similar, which is true I guess. Then why is it that the nature of graphs are different? I mean why is it that if the indifference curve is convex then utility function is quasi concave?  How did we arrive at this conclusion?

You seem to be confusing some things.

  The indifference curve and utility function both have the same
  equation

The utility function has a formula, not an equation.

$U(x_1,x_2)$ is a utility function.

The points $(x_1,x_2)$ for which $U(x_1,x_2) = 13$ form an indifference curve. (13 was picked as a random number)

The graph of a utility function has one more dimension than the graph of an indifference curve: in addition to the space of goods, this graph would also include the utility value, so its points would be something like $(x_1,x_2,u)$. Chances are you have not seen a graph of any such function, as 3D drawings are complicated. This is why maps of 2D indifference curves are used to get a feeling for the shapes of utility functions.


is there a way in NIPA accounts to see a measure of size/change in private debt stock? is there a way in NIPA accounts to see a measure of size/change in private debt stock , and how is it measured?
if not then how is it measured?

What you want is the Flow of Funds (which measures transactions and levels of assets and liabilities), not the National Income and Product Accounts, which primarily measure output and (mostly, though not entirely, physical) capital stock.


Externalities, Pigouvian Taxes and Wikipedia It states on Wikipedia:

  A Pigovian tax (also called Pigouvian tax, after economist Arthur C.
  Pigou) is a tax imposed that is equal in value to the negative
  externality. The result is that the market outcome would be reduced to
  the efficient amount. A side effect is that revenue is raised for the
  government, reducing the amount of distortionary taxes that the
  government must impose elsewhere. Governments justify the use of
  Pigovian taxes saying that these taxes help the market reach an
  efficient outcome because this tax bridges the gap between marginal
  social costs and marginal private costs.

Q: When it says, "A side effect is that revenue is raised for the government, reducing the amount of distortionary taxes that the government must impose elsewhere", what does that even mean?

The Pigovian taxes are non-distortionary. For example imagine situation where government optimal spending is 100e and before Pigovian tax all 100e was raised through income tax which creates distortions on Labour market. Let’s say that after imposing Pigovian tax government gets additional 30e. Now since government needs only 100e for its optimal spending it can either choose to run 30e surplus or choose to reduce the distortionary income tax by 30e. The Wikipedia sentence refers to this second option where government chooses to reduce some other distortionary tax thanks to the revenue from Pigovian tax.


List of production functions that satisfy the Inada conditions It is known that in the class of CES production functions, only the Cobb-Douglas production function satisfies the Inada conditions. Which other functions satisfy the Inada conditions?

The Wikipedia statement should come with a citation or proof, as it is not entirely accurate: Cobb-Douglas functions with increasing returns to scale do not necessarily satisfy the Inada conditions, e.g., $f(x_1,x_2) = x_1^2x_2^2$ does not.

There are many non Cobb-Douglas functions that satisfy the Inada conditions, albeit they may not have nice compact formulas. E.g.,
$$
f(x) = \left\{\begin{array}{ll}
x^{1/2} & \mbox{ if } x \leq 1 \\
2x^{1/4} - 1 & \mbox{ if } 1 < x
\end{array}\right.
$$
This satisfies the conditions:

$f(0) = 0$
The Hessian is negative semidefinite as $f$ is strictly concave.
$$\lim_{x \to 0}\frac{\text{d} f(x)}{\text{d} x} = \infty$$
$$\lim_{x \to \infty} \frac{\text{d} f(x)}{\text{d} x} = 0$$

You can easily manufacture similar functions. I just combined two strictly concave functions at a point of tangency and made sure that the piece going to zero would cross the origin.

I am not sure, but it seems to me that if you have an Inada satisfying function $h(\mathbf{x})$ then the function $f(h(\mathbf{x}))$ will also satisfy all the conditions.


Wooldridge's Microeconomics book vs Cameron and Trivedi's textbook I'm debating between 2 Microeconometric classes at my college, (I realize I'm fortunate to have a choice). One class uses Wooldridge's "Econometric Analysis of Cross Section and Panel Data" the other uses Cameron and Trivedi' book "Microeconometrics: Methods and Applications".

The question is, will I learn mostly the same content from both of these book or is one much better than the other?

P.S. I've been told the class using Wooldridge's book is much harder which is somewhat scary and intriguing at the sametime. Tuition is a lot of money and I don't want to take a class that is more advanced that I can handle then fail/drop. However, I want to be a good economist and be very knowledgeable on all things econometrics.

Both books are very good, and they are not competitive but rather complementary. I would say that compared to $CT$, $W$ is a more "traditional" econometrics book, while $CT$ includes whole chapters in broader underlying statistical issues. At the same time, $CT$  are relatively more concerned with the dirty reality of doing applied econometrics with real-world data sets, and how this feeds back to the theoretical models.

Regarding your choice, it matters what each instructor does with each book. Ask them.

Learn which chapters are covered in each class and compare, to see if they cover the same subjects or not... there must be a reason why there exist two available classes.


Can banks buy stocks? Can a bank use its excess reserves to purchase stocks? If so, does this inflate the cost of borrowing for your average Joe since he has to offer returns (interest payments) that are competitive with stock market returns? Or, on the other hand, maybe a personal loan with collateral is competitive when you consider the low risk?

1st of all, banking regulation in different, in various countries, so there is no 1 accurate answer for all cases.

Having that said, In general, banks may buy stocks, but that won't be with any capital being held as a deposit reserve ratio, or Basel regulation recommendations.
This is because a bank has to keep enough capital allocated per loan.
Furthermore, when a bank recruits capital in order to lend it, it becomes its own debt, rather than capital. The bank lending money is its asset, which should be alternative to stock buying

When a bank buys a stock, its done from it Nostro account, dedicated for investments and is seperate from its "classic" activity

Of course diversions from the theoretical model do exist, in cases like the Sub-prime bubble of 2008. but, obviously,the large banks back then, in fully aware manner, broke their own business model rules, in a fraudulent way


Strictly increasing function transformation I have utility function given by:

$U(x_1, x_2) =
  \begin{cases}
                                   x_1+x_2 & \text{if $x_1+x_2<6$} \\
                                   6 & \text{if $6\leq x_1+x_2\leq10$} \\
                                   x_1 + x_2 & \text{if $x_1+x_2>10$}
  \end{cases}$

If I transform this to this:

$F(U(x_1, x_2)) =
  \begin{cases}
                                   U(x_1, x_2)+4 & \text{if $U(x_1, x_2)<6$} \\
                                   10 & \text{if $U(x_1, x_2)=6$} \\
                                   U(x_1, x_2) & \text{if $U(x_1, x_2)>10$}
  \end{cases}$

Is it strictly increasing function transformation?

Thanks guys

This is a strictly increasing monotonic transformation simply because the preferences in the previous function are the same in the transformed function as well.


Definition of subgame perfect Nash equilibrium Take a two-stage game with complete information and simultaneous actions in each state:

(1) Player 1 and 2 simultaneously choose action $a_1\in A_1$  and
$a_2\in A_2$ respectively.

(2) Player 1 and 2 observe the outcome of the 1st stage
$(a_1, a_2)$, then simultaneously choose action  $a_3\in A_3$ and
$a_4\in A_4$ respectively.

Payoffs are $u_i(a_1, a_2, a_3, a_4)$ for $i = 1,2$.

As equilibrium concept I use subgame perfect Nash equilibrium. I find it by
backward induction:

(A) find the functions $a^*_3(\cdot)$  and $a^*_4(\cdot)$ such that $\forall (a_1,a_2)\in A_1\times A_2$

$$
\begin{cases}
a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
\end{cases}
$$

(B) find $(a_1^*, a_2^*)\in A_1\times A_2$ such that
$$
\begin{cases}
a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
\end{cases}
$$

Question: a subgame perfect Nash equilibrium is
$$
\{a_1^*, a_2^*, \underbrace{a^*_3(\cdot), a^*_4(\cdot)}_{\text{Functions}}\}
$$
or
$$
\{a_1^*, a_2^*, \underbrace{a^*_3(a_1^*), a^*_4(a_2^*)}_{\text{Point in the image set of the functions $a^*_3(\cdot), a^*_4(\cdot)$}}\}
$$
?

An equilibrium consists of a profile of strategies, which specifies an action for every player at each possible contingency. Since each action profile $(a_1,a_2)$ is a contingency, the SPE must include functions $a^*_3(a_1,a_2), a^*_4(a_1,a_2)$ that specifies what to do at those contingencies.


Are there two different meanings of the discount rate? Today I suddenly became aware of how the term "discount rate" are frequently used in two different contexts:

When people prefer to have something now rather than in the future. The reason is supposedly human psychology. The discount rate in this case is inherent to the individual.
The same amount of money now is more valuable than in the future. The reason is that if one has that money now, he can invest and earn interest. The discount rate in this case is the real interest rate, determined by the market.

Am I correct in thinking that these are essentially two distinct phenomena subsumed under the same term? Or there is actually a deeper connection between them?

It is standard approach by now to acknowledge the existence of a "rate of pure time preference", denoted usually by $\rho$, that characterizes individuals. This is a fundamental aspect of preferences - a "primitive" parameter. It is not a proxy for the existence of uncertainty (this is why it is found also in deterministic models), neither does it reflect "economic opportunity costs" (this is why it discounts future utility also).

In continuous-time model this discount factor takes the form $e^{-\rho t}$, while in discrete-time models, this discount factor takes the form $\beta = 1/(1+\rho)$.

In the individual's intertemporal maximization problem, the interest rate does not enter in the utility/objective function, but in the law of motion of personal wealth. So a negative real interest rate does not reflect on the personal discount rate.

Traditionally also, firms are considered to discount the future only with respect to the opportunity cost which in turn is assumed to be reflected in the (real) interest rate $r$. So when modelling the intertemporal problem of a firm we usually see $e^{-r t}$ and $\beta = 1/(1+ r)$.


How was the CES production function derived? The Constant Elasticity of Substitution production function is defined as:
(Taken from Wikipedia)
$$Q=F \boldsymbol{\cdot}\left(a\boldsymbol{\cdot}K^r+(1-a)\boldsymbol{\cdot}L^r \right)^{1\over{r}}$$
Where:
$Q=$ the quantity of output
$F=$ factor productivity
$a=$ Share parameter (i.e. $0 < a <1$)
$K, L=$ Quantities of production factors
$r= {\left(s-1 \right)\over{s}}$
$s= {1\over{1-r}}=$ Elasticity of subsitution
My question:
Though this is a quite elegant formula, how is it derived?

The CES function can be derived directly from the condition of constant elasticity of substitution.  There are various ways to do this, but the simplest derivation occurs for a homothetic production function.  Suppose we start with a homothetic production function $Q = f^*(K, L)$ and we rewrite this in intensive form as:

$$\begin{matrix} q = f(k) & & q \equiv Q/L & & k \equiv K/L. \end{matrix}$$

For this case the elasticity of substitution $s$ can be shown to be:

$$s = - \frac{f'(k)(f(k) - kf'(k))}{kf(k)f''(k)}.$$

Letting $r \equiv (s-1)/s$ and re-arranging this equation gives the second-order differential equation:

$$\frac{kf(k)f''(k)}{1-r} + f'(k)(f(k) - kf'(k)) = 0.$$

This equation has general solution $q = f(k) =  c_0 (1 + c_1 k^r)^{1/r}$ where $c_0$ and $c_1$ are constants.  Parameterising with $a \equiv c_1$ and $F \equiv c_0 \cdot c_1^{1/r}$ and substituting to obtain the extensive form gives:

$$\begin{equation} \begin{aligned}
Q = Lq = Lf(K/L) &= c_0 L \left( \left( 1 + c_1 \frac{K}{L} \right)^r \right)^{1/r} \\
&= c_0 \left(c_1 K^r + L^r \right)^{1/r} \\
&= F \left(a K^r + (1-a) L^r \right)^{1/r}.
\end{aligned} \end{equation}$$

The parameter $a$ can be interpreted as the capital intensity in production and the parameter $F$ can be interpreted as the overall efficiency of production.


How to maximize revenue by changing price? I run a SaaS and I have the following data from it:
Number of trials (A), number of subscribers (B), subscribe cost (C) and total revenue (D) for every type of service my SaaS provide.
If I increase (C) it can reduce (B) and in result I can get less (D).
So, is there a way to find balanced (C) to maximize (D)?
Also I think there is some dependency between (A) and (B) that shows subscribers solvency.

Your question does not provide enough information for me to walk you through the process step-by-step using your information, but hopefully I can point you in the right direction.

You question about Price Elasticity of Demand (usually just called Elasticity of Demand). Elasticity of Demand is used to show how responsive demand is to a change in price. For example, if a $1\%$ increase in price leads to a massive drop in the quantity demanded, say 75%, then we would say that the demand is relatively elastic. If an increase in price does not really cause a drop in the quantity demanded then demand is relatively inelastic. The concept of elasticity of demand translates to your question because $price\ multiplied\ by\ quantity\ equals\ revenue$.

You mention that if you increase $(C)$, you reduce $(B)$. Yes, this is (usually) correct because the relationship between price and quantity is negative-- the higher the price, the fewer items sold. As for reducing $(D)$, this depends on the elasticity of demand. For example,

Assume the price for each subscription is $\$2$. Assume that at a price of $\$2$, you sell $1,000$ subscriptions. This gives you a revenue of $\$2,000$: $$\$2( 1,000)=$2,000$$

Elasticity demand can be calculated a few ways, I'm going to use: $$Elascitity\ of\ Demand = {\frac  {{\mathrm  {Percent\ Change\ in\ }}Q}{{\mathrm  {Percent\ Change\ in\ }}P}}$$ Where $Q$ is quantity demanded and $P$ is price.

Lets consider the following two examples:

Scenario 1:

Assume that you increase the subscription price from $\$2$ to $\$3$. Now assume that an increase in price from $\$2$ to $\$3$ decreases the total number of subscriptions sold from $1,000$ to $960$. Notice that the number of subscriptions decreased, but revenue increased from $\$2,000$ to $\$2,880$: $$\$3(960)=$2,880$$ This happened because the demand for subscriptions is considered inelastic. Price by 50% [$(3-2)/2$], and your quantity decreased by 4% [$(960-1000)/1000$]. Since the percent of the price increase increased by more than the quantity demanded decreased, you increased your revenue. Your elasticity of demand in this scenario is, $$\frac {\mathrm -0.04}{\mathrm 0.50}=-0.08$$

Scenario 2:

Assume that you apply the same price increase from $\$2$ to $\$3$ per subscription, but this time the quantity sold drops from $1,000$ to $400$. Your revenue is now, $\$1,200$ $$\$3(400)=$1,200$$ In this case an increase in price caused your revenue to decrease. What happened? In this case subscriptions are considered elastic. The price increased by 50% [$(3-2)/2$], and your subscriptions demanded decreased by 60% [$(960-1000)/1000$].Your elasticity of demand in this scenario is, $$\frac {\mathrm -0.6}{\mathrm 0.50}=-1.5$$

To answer your question, you are looking for the place on your demand curve where the % change in price is the same as the percent change in quantity demanded. This is point is called Unit Elastic. In order to calculate this for yourself, you will need to know the relationship between the price of subscriptions and the quantity demanded.

Consider reading more about elasticity of demand as there are a couple of ways to calculate it. When you do the calculations you will get an elasticity of demand number that ranges from $0$ to $-∞$. (Some people will convert these numbers to absolute value... $0$ to $∞$, means the same thing.) The number you get from calculating your elasticity of demand will tell you what to do with your price. For example, if you get an elasticity of demand number that is greater than $-1$ (or less than $1$ if you use absolute value), then increase your price because revenue should increase. If you get back an elasticity of demand number less than $-1$ (or greater than $1$ if you use absolute value) then decrease your price and your revenue should increase. You want to adjust your price until your elasticity of demand number is $-1$ (or $1$ if you use absolute value). $-1$ is the point of unit elastic.


How come the CBO reported the Federal Deficit for FY2014 was \$483 Billion when debt went up by \$1.1 trillion? The Monthly Budget Review: Summary for Fiscal Year 2014 states that the deficit was \$483 billion.

Treasury Direct shows Total Public Debt Outstanding increasing over the same period from \$16.7 trillion to \$17.8 trillion.

Why is there such a huge difference in these figures?

First off, you are looking at the wrong column to compare like with like.

Total debt went from \$16.7tr to \$17.8tr but that includes intragovernmental debt.  Changes in this number will net out in the deficit calculation (this is one part of the government lending money to another part).

Secondly, you are starting on the wrong date.  You need to go full year, so start from 30-Sep-2013 and go to 30-Sep-2014 (you would think 1 day wouldn't matter, but a lot happens on that last day of the fiscal year!), so your link should be: Treasury Direct FY 2014

So the real number you expect to see change in line with deficit is the "held by public".

This changed from \$11.976tr to \$12.784tr, an increase of $808bn over the period.

So that leaves a gap of $325bn to be explained.

This can be explained by the fact that various things are considered "off budget" for purposes of calculating the deficit, a small example I can remember is the U.S. Postal Service.  Prior to Obama the Iraq and Afghanistan war were considered "off budget".  I think various overseas aid payments are also off-budget.  Also any loans or asset purchases or financial gains and losses will impact the debt value, but will be treated "off budget".

This "off-budget" idea may seem a little dodgy but it should probably be seen in comparison to corporate accounting.  For corporate accounts you would not expect the income statement and cash flow statement to exactly match.

Doing the Math for 2013

Ok, so this is the best I can find, I'm afraid it is projected for 2014 but it gives you the idea.  We can do the calculation for 2013.

The Budget for Fiscal Year 2015

Go to page 207, then there is a table showing the bits that contribute to the debt change.

If we re-ask your question for 2013 (where we have data), then your links are:
Monthly Budget Review: Summary for Fiscal Year 2014
and
Treasury Direct FY2013

Then we see that the deficit was \$680bn and the public debt changed from \$11,269,585,800,039.32   to \$11,976,279,236,073.83 = \$706.7bn

Now according to the budget tables, we see that Unified Deficit is \$680bn but that there is an additional 22bn of changes in financial assets and liabilities, bringing us up to $702bn.

Not bang on, but probably as close as you are going to get - rounding and different sample dates probably makes the difference.  You can see from the projections for 2014 that there is a much bigger value of \$271bn, which is only (!) \$54bn short of the actual number we are seeing.


Is technical analysis somehow a valid method for investment? I personally think technical analysis (TA) is not useful at all but I still see lot of courses about technical analysis, and even people using it.  They have the advantage of being easy to learn and they claim themselves as following the "behavioral economics" theory.

Is the traditional & simple TA somehow valid?  I'm asking about the one that uses moving averages, graphs, momentum lines to try to predict market moves.

The behavioral economics theory is valid and 2 people won Nobel prizes because of that, so I'm not arguing its validity, but its consequences or usefulness as an investment method.

Please take into account not only a highly efficient market (eg, the US), but other smaller and less efficient markets (I'm from Argentina).

The Lo, Mamaysky and Wang paper Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation is probably the best investigation of this question. He found there were some patterns that seemed to be useful, notably the Head and Shoulders pattern.


Zero sum game, constant sum game Given any bilateral zero-sum game G, show that strategy proﬁle σ is a Nash equilibrium for G if, and only if, it is a Nash equilibrium for the constant-sum game G' obtained from G by adding any ﬁxed amount "d" to the payoffs of both players. Is the conclusion affected if the ﬁxed amount, call it now $d_i$ for each i = 1 , 2 , differs between the two players?

Suppose $(\sigma_1^*, \sigma_2^*)$ is the Nash Equilibrium of the game $G$ consisting of players $\{1,2\}$ having strategy sets $A_1$ and $A_2$ and the payoff functions $u_1:A_1\times A_2 \rightarrow \mathbb{R}$ and $u_2:A_1\times A_2 \rightarrow \mathbb{R}$. Consider the game $G'$ in which everything else is same as $G$ except that the payoff functions are $v_1:A_1\times A_2 \rightarrow \mathbb{R}$ and $v_2:A_1\times A_2 \rightarrow \mathbb{R}$ defined as: $v_1(a_1, a_2) = u_1(a_1, a_2) + d_1$ and $v_2(a_1, a_2) = u_2(a_1, a_2) + d_2$. We will now show that $(\sigma_1^*, \sigma_2^*)$ is also the Nash Equilibrium of $G'$.

Since $(\sigma_1^*, \sigma_2^*)$ is the Nash Equilibrium of $G$,
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*)) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2)) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

The above can also be rewritten as
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*)) + d_1 \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*)) + d_1 & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*)) + d_2 \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2)) + d_2 & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

Because $d_1$ and $d_2$ are constants, we can take them inside the expectation and rewrite the above inequalities as:
\begin{eqnarray*} \mathbb{E}(u_1(\sigma_1^*, \sigma_2^*) + d_1) \geq  \mathbb{E}(u_1(\sigma_1, \sigma_2^*) + d_1) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(u_2(\sigma_1^*, \sigma_2^*) + d_2) \geq  \mathbb{E}(u_2(\sigma_1^*, \sigma_2) + d_2) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

By definition of $v_1$ and $v_2$,

\begin{eqnarray*} \mathbb{E}(v_1(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(v_1(\sigma_1, \sigma_2^*)) & & \forall \sigma_1 \in \Delta (A_1) \\ \mathbb{E}(v_2(\sigma_1^*, \sigma_2^*)) \geq  \mathbb{E}(v_2(\sigma_1^*, \sigma_2)) & & \forall \sigma_2 \in \Delta (A_2) \end{eqnarray*}

Therefore, $(\sigma_1^*, \sigma_2^*)$ is also the Nash Equilibrium of $G'$.


Why are aeroplanes not made in one place? As an example, different parts of an Airbus A380 are produced in factories all over Europe, and then assembled in a factory in France. Transporting these massive objects must be incredibly expensive (unless in fact they are shipped in small parts).

Why not build it in one place, like ships?

Short answer
Airbus is a particular case of decentralised supply chain which came from historical and political reasons. Airbus is a consortium with several European states having a stake on it, which means that each state wants a part of the production in their country.
Long answer: Why is Airbus' supply chain disperse
The aviation industry for commercial planes basically has has two large players nowadays: Boeing and Airbus. The 70's-80's saw the consolidation of American producers, with Boeing as the survivor and most successful of them (incorporating other large players such as the old McDonell Douglas).
Europe had smaller producers, such as BAC or Aerospatiale. Smaller European producers in the 70's and 80's wanted to compete with Boeing, namely with the success of the 727, but the fragmented market would be a challenge. Therefore, Airbus was a consortium created with several European manufacturers to have a large aviation player with European roots. To minimize political difficulties, the supply chain was split over several countries:

While the original partners in the company recognised that the only way to compete was to consolidate national industries on a regional European basis, consolidation required negotiation and political finesse. Most sensitive was the decision where to site the different manufacturing operations.
The easiest way to defuse tensions was to build different bits of the first plane, the A300B, in the different partner countries. The French made the cockpit, the control systems and the lower-centre section of the fuselage; the UK made the wings, and the Germans made the rest of the fuselage and a part of the centre section. The Dutch made the moving parts of the wing, the flaps and the spoilers, while the Spanish made the horizontal tailplane.

Source: Financial Times
This supply chain is unlikely to become more centralised or placed offshore, for political and economic reasons.

From the economic perspective, today's supply chain is too fixed to be able to be moved somewhere cheaper, as the regions where each part is produced developed local expertise and innovation centres around it. Another argument is that the de-centralized supply chain helps to have a faster delivery time (see FT article mentioned above).

From the political perspective, European stability (commercial and military) requires that countries depend on each other for developing aircraft. Although Airbus is mostly known from the A380s and A320s, they also produce military aircraft with their subdivision Airbus Defense and Space. Placing production outside of Europe would place aviation expertise in other countries, a strategic expertise to have for defense.


Why is investment not a cost in firms profit functions? In all textbooks I've seen profits is written as: $$\pi=py-wL-rK$$ It is evident here that w and r are operational costs.

But unless there is no depreciation and the level of capital is to remain fixed, there is a cost of investment! Why is this not taken into account for calculating profits? Is it some sort of assumption of "steady state" involved?

Both in Economics and in Accounting, there is the following fundamental principal: we have to subtract revenues generated in a given time period from costs incurred in the same time period (because this is what makes basic sense).

Now, "cost" is the value of productive resources absorbed into production, in the given time period. The total value of an "Investment" is not absorbed in the usual time period where firms record their profits (month, quarter, year).
When we buy a machine, we use it for many years.
So only a part of "Investment" is absorbed in a period (essentially what constitutes an "investment" and what an "(operating) expense" is based on this criterion exactly -whether it is fully absorbed in production in one time period or not). The total value of Investment done in say one year is an outflow in the Cash-Flow statement, but it is not a Cost in the Profit and Loss statement.

And, as you mentioned, that part of the Investment that was absorbed in production during the period, is reflected in the calculated Depreciation, which is chraged as a cost to the Profit and Loss statement.


How would a foreign trade multiplier behave if imports depended only on exchange rates? I was wandering if the amount of imports and exports in an economy depended only on the exchange rates then what would the foreign trade multiplier of this economy look like ?

If the net exports depended only on the exchange rate then there would not be any induced effect of net exports on income (as exports would not depend on income) hence, no multiplier effect. The other multipliers in economy would not be affected either.


Determining the Gini Coefficient in a 2 person economy In a 2-person economy, the owner of a firm pays 20% of the total output to the worker.

What is the Gini coefficient in this economy?

The answer is supposed to be 3/5 but I end up getting 3/10 instead.

Gini Coefficient is the ratio of the area that lies between the 45 degree line (line of equality) and the Lorenz curve, and the area below the line of equality. Given the data in the problem Lorenz curve connects points $(0, 0)$, $(50, 20)$ and $(100, 100)$ in the graph. To find the Gini coefficient, we find the area between the line of equality and the Lorenz curve and divide it by the area of the triangular region that lies below the line of equality.

Gini coefficient $\displaystyle = 1 - \frac{\color{blue}{0.5\times 50 \times 20} + \color{red}{(100 - 50) \times 20} + \color{gray}{0.5\times (100-50) \times (100-20)}}{0.5\times 100 \times 100} = 0.3$

Here is the graph:


Can the endowment point be the competitive equilibrium? Quoting from MWG p.519, Chapter 10: Pure Exchange, The Edgeworth Box

  ..., any intersection of the consumers' offer curves at an allocation different from the endowment point $\omega$ corresponds to an equilibrium..."

Does this mean that the endowment point can never be an equilibrium or that the crossing of offer curves there is not enough to guarantee that?

The endowment can be an equilibrium allocation. But as is written on page 518, if indifference curves are smooth, which we will assume in the following, they must be tangent to the offer curve at the endowment. So if the offer curves cross at the endowment, the indifference curves must cross too. But if the indifference curves at the endowment cross each other, the endowment cannot be Pareto efficient. But every equilibrium allocation is efficient, so the initial endowment will not be an equilibrium allocation if the offer curves cross each other there (they would need to be tangent.)


Game Theory of Two Firms There are only two firms that produce and sell hockey equipment, Abel Corporation (A) and Baker Company (B). They each sponsor special events to try to gain customers. As there are only 2 firms, the marketing of one firm clearly affects its own profits and the profits of the other firm too.

Here are the (Gross) Profit Functions of each firm, related to the number of special events in a year, where A represents the number of events that Abel undertakes and В represents the number of events that Baker undertakes.

• Profits, Abel = 1000A -A^2 - 0.5AB

• Profits, Baker = 1000B - B^2 - 0.5ВA

Each special event costs $160, which is NOT included in the Profit functions as shown above.

What is Abel's Reaction Schedule (Best-Response Function)?

Here's how I've done this, I'm not sure if it's right, that's why I'd like your feedback.

Abel's Reaction Function:

1000 - 2A - 0.5B = 160   (differentiated the profits function and equated to 160)

A = 420 - 0.25B  (this is the reaction function I derived)

My confusion arises from the thought process that the profits function is essentially the difference between total revenue and total costs so should that be something i need to take into consideration when solving this?

Your answer is correct. Given that you have a net profit function once you substract the costs of the events, the way to proceed is differentiating that profit function, taking the behaviour of the other firm as fixed and setting that to 0.

Profit maximization involves producing upto the point where marginal benefits equal marginal costs and that is exactly what you do by differentiating and setting to 0.


Why does the Fed feel the need to reduce its balance sheet? Why does the Fed feel the need to reduce its balance sheet?

What is the problem with the Fed having a large balance sheet long term? What would happen if the value on the Fed's balance sheet collapsed, say in a bond crash?

This is veering into the direction of an opinion-based answer, but I would argue that there is no particular evidence that the Fed “needs” to reduce the size of its balance sheet.

If the Fed wanted to revert to its pre-2008 mode of operations - where interest is not paid on reserves - it would be required to reduce its balance sheet so that there are no excess reserves in the system, if it wanted the Fed Funds rate to be greater than zero. In the aftermath of the Financial Crisis, many commentators  assumed that this was the case. However, the pre-2008 Fed operating procedure was unusual relative to what other central banks do, and there is no “need” to revert to it. By paying interest on reserves, it can place the overnight rate wherever it wishes (with small spreads).
The Fed does not mark-to-market its portfolio. Market value losses on bonds does not present an issue, as they can hold to maturity. The only “risk” is that the cost of reserves is higher than coupons received. However, like all central banks, it can operate with negative equity. At most, negative equity is a political embarrassment.
Are there economic effects of a large balance sheet that need to be undone? This is a disputed area of research. My opinion is that the side arguing that the size of the balance sheet matters have not presented strong evidence of this claim. One would need to look at claims about the effects of the balance sheet growth, then see whether they hold up. From what I saw of the post-Crisis literature, the effects were not clear-cut.


The optimal price for a demand curve with a steep slope Given the demand function,
$$D(p)=A-ap$$
I've found the optimal price,
$$p=\frac{A+ac}{2a}$$
Where $c$ is cost and $A,a >0$.
My question is how is the optimal price is dependent of $a$ (1) - what will happen with the optimal price when $a$ is arbitrarily high? (2)
(1): My answer is that the optimal price will decrease with an increasing $a$. But what happens when $a$ is arbitrarily high?
(2): My guess is that the optimal price will near $\frac{c}{2}$, but I can't put  a word on why this happens. Why would the firm take an optimal price that is lower than the cost?
It doesn't seem very intuitive for me.

That portion of answer about decreasing price is not generally correct as a is both in numerator and denominator and you don't know what $c$ is. You can answer the question about what happens to $p$ as $a$ becomes arbitrary high using limits. Taking your solution for optimum price and taking limit as $a$ goes to infinity gives:

$$\lim_{a \to \infty} \left(\frac{A+ac}{2a}\right)=\frac{c}{2}$$
(here we had to use L'Hôpital's rule as the limit was in indeterminate form of $\infty/\infty$). So if $a$ is arbitrarily high the price will be half of $c$

You do not mention the whole problem is so it is hard to be precise here (you say $c$ is cost but $c$ is usually marginal costs in these problems), regardless optimum price can be below costs that would just mean that firm would eventually exit the market and stop providing the service. That is because the demand is simply too much decreased by even small increases in price.
This is essentially situation where people would want some goods or service but only if price is so low that firms are not willing to provide, so the market won't exist. This can happen in reality. You probably had at least once in your life thought that it would be nice if there would be movie/game/book/some service etc that caters to some niche taste but you see that nobody is providing it. Well one of the reasons why nobody is providing it might be that there is no demand at the price where it would actually be economically viable to provide the good or service.


What would happen in a modern economy if growth plateaus? According to a Wikipedia article on a United Nations report,

The UN Population Division report of 2019 projects world population to continue growing, although at a steadily decreasing rate, and to reach 10.9 billion in 2100 with a growth rate at that time of close to zero.

Aiming for a low to zero or even negative growth rate for the world population makes sense, considering the limitations on living space and resources on the planet (though if the likes of Jeff Bezos and Elon Musk have their way, there will be expansion into outer space, creating new vistas of living space and resources). If the growth rate for the population is low, it would take considerable effort to maintain existing growth rates in the economy, since the number of people available to produce capital would be growing at a lower rate or not growing at all.
This inspires the question: If the growth rate of a modern economy, such as that of the United States or Canada, plateaus (meaning the amount of capital in the economy does not grow or grows very little), what would be the ramifications in the nation and internationally?
Some factors to consider:

Private and public retirement funds are invested in the economy and require growth in order to be able to pay its aging members.
Would fractional banking and the credit system live if there was zero to low growth for a long enough period of time?
What effect would there be on inflation?

I'm sure that there are many other factors other than these three to consider. Any thoughts would be appreciated. I have seen one post and another post, but they are more about "Why is growth good?" whereas I am wondering "What would happen to our existing system if we failed to have sustained exponential growth?"

Caveat
 Your main question is simply too broad to cover in SE format so this answer will focus only on the 3 further sub-questions.
tl;dr
The premise of the question is actually not generally valid. Economic growth is not driven primarily by population growth, especially not per capita economic growth (which is what matters for living standards). However, suppose that there will not be any economic growth for some other reason. The short answers to your 3 questions would be:
A1: This would put stress on the pension system. However, It would not make the pension system unworkable as pensions can work on the same principle as an insurance and that does not require economic growth for it to be viable. Nonetheless, at the same time it would make pension schemes more expensive which would mean people would have to either; a) work longer; b) have lower pensions; c) payments into these pension plans (or taxes in case of government) would have to increase.
A2: I can't see any economic reason why this would affect fractional reserve monetary systems in any way whatsoever. In fact this won't be further addressed in full answer because there is not much more to say about it. Fractional reserve system simply works equally in a falling, stagnant or growing economy.
A3: This would not have any effect on inflation in itself ceteris paribus, but there could be some effects caused by the way the government chooses to respond to the issue of no growth.
Full Answer
Addressing the Premise
Before addressing the questions let me address problems with your premise. You argue that:

If the growth rate for the population is low, it would take considerable effort to maintain existing growth rates in the economy, since the number of people available to produce capital would be growing at a lower rate or not growing at all.

This is simply not a generally accepted view in growth literature.
First, what matters for the economy is growth per capita (per person) not just economic growth in general. For example, if GDP doubles from 100 to 200 but also the number of people increases from 1 to 2, GDP itself grows but per capita GDP in both scenarios is just 100. It is growth per capita that matters for material welfare of people  (see discussion of living standards in Mankiw Macroeconomics or Blanchard et al. Macroeconomics: a European Perspective).
Second, most economists don't believe that per capita growth has much to do with population growth. In economics there are two main strands of literature when it comes to economic growth. There are exogenous theories of economic growth and endogenous theories of economic growth. Both of the above mentioned strands of literature agree that per capita economic growth is caused by increase in technology but they disagree on what can cause increase in technology. In the exogenous growth literature (which is actually more dominant) population growth has no effect on GDP per capita whatsoever. In the literature on endogenous growth it might have an effect on GDP per capita through R&D (more people means more scientists) but even most endogenous growth models what matters more than population is investment in R&D and share of population devoted to producing R&D. Even in the small subset of models zero population growth would not lead to zero growth per capita - only to lower growth than otherwise (see discussion in Ch 1 and 3 in Romer Advanced Macroeconomics and Barro & Sala-i-Martin Economic Growth). This being said for the rest of the answer we can just assume zero economic growth as a thought experiment.
Answer to question 1 on pension systems
Zero economic growth and population growth would definitely put a stress on pension systems but they do not require economic growth or population growth to operate.
Generally speaking pension schemes work on a similar principle as insurance. In fact you can consider it an insurance for reaching a certain age. Such a scheme does not require growth in population or economic growth per se. However, economic and population growth make these schemes cheaper for the government (respectively people in case of private pension schemes).
Population growth makes  pension schemes cheaper because it improves the dependency ratio (i.e. how many working age people support one retiree - see further explanation at Eurostat). However, dependency ratio is affected by large number of things that are completely in control of government (company offering pension). For example, retirement age directly affects dependency ratio because people at some point die from old age. Thus, in principle if you set retirement age high enough you can get as favorable dependency ratio as you want.
Next the economic growth helps in the case of government pay as you go schemes because it increases peoples income and thus indirectly taxes that government collects (20% tax from €1000 is higher than 20% tax from €100). Hence as government budget grows over time these pensions can become more generous. In absence of any economic growth government would just have to face tighter constraint on its spending. In the case of private insurance where retirement money are invested an additional problem would be that if there is no growth investment opportunities will be low to none. However, this just means that money people save for retirement do not accrue interest. But pensions can still exist they will just not be such a good deal for people anymore as before.
If there would be no population and economic growth governments (people) could in principle adjust to the situation by:

Increasing the retirement age. This automatically improves the dependency ratio as if there is less people who are eligible
Stop increasing pensions over time.
Increase taxes (payments with private schemes) to raise more resources for such schemes.
Make pensions less generous.

The above might be serious political issue but from economic perspective it is completely feasible. Political problems are for political scientists to worry about.
Answer to question 3 about effect on inflation
There does not need to be any necessarily.  Inflation is determined by changes in price level which in turn depends on what the money market equilibrium is. This is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY \implies
\ln P=\ln M+ \ln V− \ln Y $$
Where M is the money supply, V velocity of money, P price level and Y output. In the log-linearized version the effects of right hand side variables can be interpreted as % changes. If there is  zero economic growth that  means $\Delta Y = 0$, which holding everything else constant would have no effect on price level.
When it comes to the other variables $M$ is under control of the government. Hence, it can stay put in principle. One could argue that due to lack of growth government might face higher pressures on monetary financing (due to the problems discussed in previous answer for example). But this would not be caused by the lack of growth itself.
Lastly, $V$ is historically very stable, although it does depend on interest rate. To be more specific generally when interest rate decreases velocity also decreases. To the extend that zero growth would cause drop in interest rate it would lead to deflation equal to the rate at which velocity dropped. However, velocity cannot drop forever (if velocity is 0 that means money are not used in economy at all). So there would be temporary deflation, which could be offset by government  increasing money supply proportionally as to keep prices constant.


Consumer surplus in case of perfectly inelastic demand How do we define consumer surplus in the case of perfectly inelastic demand?

This question was inspired by the comments following this answer. For a graph of inelastic demand please also see the linked question.

A motivation for a definition would be that while consumer surplus would be strange in this special case, change in consumer surplus may still be measurable easily with quite reasonable definitions.

From a purely theoretical perspective, if an individual's demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.

In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.


Calculate the substitution effect Utility function

$$
U=4{q_1}^{0.5}+q_2 \\
Y=10
$$

$$
p_1=p_2=1
$$

Then $p_1$ rises to 2. I would like to ask how to calculate the substitution effect on the demand for q1?

What I have done is calculate the optimal bundle initially which is (4,6) then subs into the utility function and get $U=14$. However, I am stuck here for calculating the subs effect on $q1$.

You made the right start by calculating the utiltiy at $p_1=p_2=1$.

When we change a price, two things happen. Firstly, one of the goods becomes relatively more expensive, so people substitute away from that good.

Secondly, since the total amount of goods someone can afford is lower when a price increases, it is as if their income went down.

To find the substitution effect, we need to shut down the second of these effects and focus on the first.

It sounds like you are after what is more properly known as the Hicksian substitution effect. To calculate that, we need to compensate the consumer for the aparent loss of income. In other words, we need to answer the following question: "Given $p_1=2$ and $p_2=1$, what income would be needed to achieve the same utility as before (i.e. $U=14$)?" Call the income level that answers this question $\widetilde{Y}$.

Then, all we need to do is calculate the optimal consumption bundle when $p_1=2$, $p_2=1$, and income is $\widetilde{Y}$. The substitution effect is the the difference in $q_1,q_2$ between this new bundle and the first one you calculated.

Hopefully, this is enough to help you figure out the solution.


Karush-Kuhn-Tucker in infinite dimension Does the Karush-Kuhn-Tucker theorem on sufficient conditions for optimality of a convex program apply in countable dimension?

For precisions, see Definition 4.1.1 and Theorem 4.1.4 of this course. Does the theorem extend to an infinite (but discrete) number of variables and associated constraints? If so, do you have a reference?

Yes, Bachir et al. (2021) extend the Karush-Kuhn-Tucker theorem under mild hypotheses, for an infinite number of variables (their Corollary 4.1).
I give hereafter a weaker version of the generalization of Karush-Kuh-Tucker in infinite dimension:

Let $X\subset\mathbb{R}^{\mathbb{N}}$ be a nonempty convex subset of
$\mathbb{R}^{\mathbb{N}}$ and let $x^{*}\in Int\left(X\right)$. Let
$f,g_{1},g_{2},...,g_{m}:X\rightarrow\mathbb{R}$ be convex functions
continuous at $x^{*}$ and term-to-term differentiable at $x^{*}$, i.e.
such that the functions
$f_{n,x^{*}}\left(x_{n}\right):=f\left((x_1^{*},...,x_{n-1}^*,x_n,x_{n+1}^*,...)\right)$ and
$g_{j,n,x^{*}}\left(x_{n}\right):=g_{j}\left((x_1^{*},...,x_{n-1}^*,x_n,x_{n+1}^*,...)\right)$ are
differentiable at $x_{n}$ for all $n\in\mathbb{N}$ and
$j\in\left\{ 1,2,...,m\right\}$.
(Qualification condition) Suppose that for all $k\in\mathbb{N}^{*}$
and for all $x\in X$,
$$x^{*}+P^{k}\left(x-x^{*}\right)=\left(x_{1},...,x_{k},x_{k+1}^{*},x_{k+2}^{*},...\right)\in X$$
If there exist
$\left(\lambda_{j}^{*}\right)_{j}\in\left(\mathbb{R}_{+}\right)^{\mathbb{N}}$
such that
$$\lambda_{j}^{*}g_{j}\left(x^{*}\right)  =0,\:\forall j\in\left\{
1,2,...,m\right\} \quad \quad \quad \quad \quad (1)$$
$$f_{n,x^{*}}^{\prime}\left(x_{n}^{*}\right)+\sum_{j=1}^{m}   \lambda_{j}^{*}g_{j,n,x^{*}}^{\prime}\left(x_{n}^{*}\right)=0,\:\forall
n\in\mathbb{N} \quad \quad (2)$$
(Sufficiency) Then $x^{*}$ is an optimal solution on $\Gamma:=\left\{
\left(x_{i}\right)_{i}\in
X\,:\,g_{1}\left(x\right)\leq0,...,g_{m}\left(x\right)\leq0\right\} :$
$$f\left(x^{*}\right)=\underset{x\in\Gamma}{\inf}f\left(x\right)$$
(Necessity) Besides, if $x^{*}$ is an optimal solution on $\Gamma$ and
if the Slater condition $Int\left(\Gamma\right)\neq\emptyset$ is
verified, then there exist unique
$\left(\lambda_{j}^{*}\right)_{j}\in\left(\mathbb{R}_{+}\right)^{\mathbb{N}}$
which verify the (Karush-Kuhn-Tucker) conditions (1) and (2).

The number of constraints has to be finite, but simple constraints like non-negativity constraints can be replaced by an equivalent restriction on the domain of the variables. For example, instead of the constraints $\forall n \in \mathbb{N},\;x_n \geq 0$ on the domain $\mathbb{R}^{\mathbb{N}}$, one can take $X=(\mathbb{R}_+)^{\mathbb{N}}$, and the theorem applies.
Note that the (sufficiency) result is easy to prove when one further assumes that the convex Lagrangian $\mathcal L(x, \lambda)=f(x)+\sum_{j=1}^m\lambda_j g_j(x)$ is Gateaux differentiable, with a Gateaux derivative equal to 0 at $u=(x^*, \lambda^*)$.
Indeed, a function $h: V \rightarrow \mathbb{R} $ convex and Gateaux differentiable on $V$ verifies $h(v)-h(u) \geq h^\prime(u; v-u), \forall u,v \in V$, where $h^\prime(u; v)$ is the directional derivative of $h$ at $u$ in the direction $v$. (One can see that from the definition of convexity: $h(u)+\theta \left( h(v) -h(u) \right) \geq h\left(u+\theta (v-u)\right)$; subtracting $h(u)$, dividing by $\theta$, and taking the limit when $\theta \rightarrow 0^+$; see this for more details). Applying that inequality to the Lagrangian at $u$ proves that the Lagrangian admits a minimum at $u$, which solves the minimization program: $ f(x^*) =L(x^*, \lambda^*) \leq f(x)+\sum_{j=1}^m\lambda_j g_j(x) \leq f(x), \; \forall x\in \Gamma$.
However, in general, it is not easy to prove that the Gateaux derivative of a convex series (such as an infinite Lagragian) (exists and) equals 0 at some point $u$, unless one uses the result (Theorem 3.14 in Bachir et al. (2021)) that the Gateaux derivative is thus equal to the sum of derivatives of each term in the series.


How many major economic systems are there? One of the biggest political debates regards the best type of economic system, which usually boils down to capitalism vs socialism.

Are there other kinds of economic systems, aside from more "primitive" kinds, like barter?

To put my question in perspective, I think of economic systems as a continuum that looks something like this:

communism | socialism | "mixed economy" | capitalism | "free market capitalism" (anything goes, rampant corruption and inequality)

So "mixed economy" would be a combination of socialism and capitalism, rather than a distinct economic system.

But that's just my opinion. There may be views that are radically different than mine

Wikipedia has a comprehensive list of  economic systems.
If you want to start thinking about number major economic systems that exist I recomend looking at the section on an economy's  allocation mechanism.
In particular, Wikipedia lists the following kinds of system:

The basic and general economic systems segmented by allocation are:

Market economy ("hands off" systems, such as laissez-faire capitalism)
Mixed economy (a hybrid that blends some aspects of both market and planned economies)
Planned economy ("hands on" systems, such as state socialism, also known as "command economy" when referring to the Soviet model)
Traditional economy (a generic term for older economic systems)
Participatory economics (a system where the production and distribution of goods is guided by public participation)
Gift economy (where an exchange is made without any explicit agreement for immediate or future rewards)
Barter economy (where goods and services are directly exchanged for other goods or services)
Post-scarcity economy (a hypothetical form where resources aren't scarce, such as Marx's concept of a Communist society)


China causing Currency War Recently there have been reports of China devaluating its currency against the U.S Dollar. I'm a layman and I am unable to understand the need for a country  devaluating  it's own currency. How would it help that country? Moreover newspapers report that this move by China could result in a currency war. So what exactly is a currency war? Please try to keep it simple, as I'm a layman in the field of economics.

Okay, in layman's words, devaluing yuan (against USD) means, Chinese made goods are cheaper in USD. On the other hand, American made goods are expensive in China as a result. China wants to keep its export competitiveness and may want to keep its 'world's factory' status. Well, South Korea may as well want to use a similar strategy to make domestically produced export goods competitive, similarly, other aisan governments considering devaluing their currencies. There would then be a currency war... hope this is as simple as you wanted to see.


Does Craigslist Reduce Unemployment? If a substantial (though admittedly I have no idea how large) part of the natural rate of unemployment is caused by the job search, do more efficient means of communication create more efficient labor markets, and if so, can anyone quantify their impact with data?

Kroft and Pope (working paper, published in JoLE 2014) ask exactly this question, and their tentative answer is "no".

They view Craigslist as a unique opportunity to study the benefits of online job sites, since it grew rapidly and somewhat idiosyncratically while other popular sites grew steadily (leaving few opportunities for identification). That said, they don't have a great identification strategy - they just treat the rise of Craigslist starting in 2005 as an event study and compare MSAs where it grew quickly to MSAs where it grew sluggishly, making the case via various alternative specifications and falsification tests that there isn't any substantial bias.

Interestingly, they find that Craigslist does seem to have crowded out traditional newspaper classified ads, but had no discernible effect on unemployment. (This doesn't surprise me too much, since Craigslist seems like a pretty minor advance over traditional classifieds for job search, and if anything you'd expect a contribution from other sites that are more specialized or make more efforts to facilitate matching.)

On the other hand, they find that Craigslist does seem to have some effect on rental vacancy rates. Perhaps simple, searchable Craigslist-style classified ads facilitate rental search (I know they have for me) but are too crude to do much for employment.


Can the growth rate increase after a change in the base year? Nowadays, I'm observing that the government of India is predicting much higher GDP growth for the Indian economy. I hear that the government has changed the base year, such that it is able to show a high growth rates.

Is that true?
Can anyone explain how it works?

I don't know what the Indian government's growth accounting practices are, but the World Bank has an independent forecast that shows India's GDP growth increasing in the near-term (pink area is a forecast):

The source is here: http://data.worldbank.org/country/india, where you can get more detailed data.


Modelling Bounded Utility Functions I'm trying to work out how to model a utility function that is bounded below some level. More precisely, given a specified limit $L$, I want to work out how to ensure that the utility of any outcome $o$ asymptotically approaches $L$ as the value of $o$ increases.

I'd really appreciate any help offered!

There are many such utility functions. Most commonly:

\begin{equation}
u(x)=L-\mathrm e^{-ax},\tag{1}
\end{equation}
where typically $a>0$. Or
\begin{equation}
u(x)=L-(x-a)^2.\tag{2}
\end{equation}

Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.


Accounting for household work in the GDP It is often told that domestic work is not included in the GDP estimates due to difficulty in measuring them. I wanted to know if there are any advances in measuring this, or any alternatives that have been devised or any other development in research in this field.

Are there techniques, regardless of practically how difficult they are, of possibly measuring domestic work?

Have such estimates been made by researchers, maybe at scales smaller than the national economy?

What are some theoretical approaches to understanding this? I remember Amartya Sen developed a model about intra-household trade. Is this related? Are there other such well known models?

Are there techniques, regardless of practically how difficult they are, of possibly measuring domestic work?

Yes, measuring household gross production (GDP measures gross product not amount of work), is just matter of estimating how much households produce, in a given unit of time (e.g. quarter, year... etc), and what would be value of that output on the market (or with income approach how much would household get if the household production was supplied to the market). Both can be estimated via various econometric techniques.
Nowadays econometrics is too broad to provide overview of all possible techniques that could be used. However, a commonality among those techniques is that they estimate the value of household production by  inferring it from value of observable output. For example, one simple method just looks at value of production that is done in market (e.g. nanny services) and then assigns the same value to the amount of that kind of output produced by household in house. Other approaches try to measure the opportunity cost of households for home production and so on. You can have look at Dulaney et al. (1992) for overview of various methods, but there is quite a lot of heterogeneity in approaches that can be applied here especially when it comes to the details.

Have such estimates been made by researchers, maybe at scales smaller than the national economy?

Well this was already done on scale of national economy. In fact, Bureau of Economic Analysis (BEA) already provides estimate of value of US household production that is estimated in a way to make it complementary to GDP (i.e. you could add it on top of it - although that does not mean that you should as it is also less precise than just GDP itself, so it depends on a particular research if that is worth while sacrifice). In fact BEA even produces household production adjusted GDP (see their article on that here). Although, most statistical offices around the world don't provide such estimates yet.
Beyond statistical offices, individual researchers were providing estimates for value of household output already back in 70s and possibly even earlier (e.g see Weinrobe, 1974 as an early example), with various degrees of precision and success. However, I don't know any study that would do that on a regional level (and quick google scholar search did not revealed any either), but as mentioned at the beginning that would be actually much harder to estimate.

What are some theoretical approaches to understanding this? I remember Amartya Sen developed a model about intra-household trade. Is this related? Are there other such well known models?

This seems unrelated to the above questions. You can estimate household output without having a theoretical understanding what determines how much output will be produced by households. This being said, in economics there is a specific subfield of economics of family. However, it is beyond scope of any reasonable SE answer to provide literature review of whole field, although generally speaking the theories of household productions are somewhat analogous to the regular ones (e.g. households will have production functions, their stocks of labor and capital, costs, home production competes with production that could be gotten from market, they will try to optimally allocate their resources to maximize utility etc.). You can have look at overview of various theories of household production in Browning & Martin  Economics of the Family or Menchik Household and Family Economics or the classic text Gary Becker Family Economics (although slightly out of date now).


labor leisure trade off Walras has available 24 hours per day. He has to alloacate this 24 hours between leisure $(L)$ and work. His utility function depends on leisure and the composite good $(C)$ and is given by $U=LC$. Work pays him a wage of 1 dollar per hour and this is his only income.

What is his optimal choice of $L$ and $C$ ?

I understand that $MRS=C/L$ and it should be equal to the price ratio. Is it 1 or 2?
I am confused whether budget constraint is $C+L=24-L$ (in this case price ratio is 2). Can someone clarify this?

My suggestion would be to take the habit of fully writing down and then solving formally even the simplest model. Yours is a static one,

$$\max_{C,L} U(C,L) = CL,\;\;\;\; s.t. C = (24-L)w$$

since we silently assume that the individual cannot consume more than his income. Here $L$ is leisure, not work.

Inserting the constraint into the objective function,

$$\max_L U[C(L), L] = (24-L)w\cdot L = 24wL - wL^2 = w(24L - L^2)$$

This permits you to see that the wage doesn't matter for the optimal allocation (it will matter, for the level of utility attained). This may sound counter-intuitive... It is, and it has to do with the category of functions to which the specific utility function belongs to. In other words, it is an unrealistic aspect of the model introduced by a mathematical choice we made.

The above has a straightforward solution for a maximum.

(PS: Has Walras been notified that his name is used in such a manner?)


Topological concepts in economic theory QUESTION: What are the major or systematic applications of post-1960s mathematics to microeconomics?

For example, in the late 19th century, Fisher first used the mathematical ideas of Gibbs to construct modern utility theory. In the 20th century, Mas-Colell incorporated topological ideas to study general equilibrium. What about the late 20th, early 21st century?

For example, consider directed graph theory, measure theory, topology, the category theory and modern homology or cohomology, topos methods, functional integration, etc.

Note 1: econometrics/statistics, without modelling, is excluded. The only modern mathematics used there is random walk theory, and the ergodic problem, solved via complex analysis. RW and EP are not specific to economics.

Any appropriate economics publication is an answer. This included also those published in non-strictly economics journals, e.g. the Journal of Mathematical Psychology.

Note 2:
Yes, I know, this type of work is rarer (not to be confused with obscurity: some of it is well known). That is what makes it easy to miss such a reference when it is published. Hence the question.

I strongly suspect that an emerging important area for applications of measure theory will be in approximate dynamic programming techniques. Approximate dynamic programming (aka "reinforcement learning" in the computer science literature) has been the direction of research work in the last ~10-20 years of the dynamic programming literature. Economics is only just now starting to adopt some of these advances. For example of the direction of the DP literature, see Bertsekas' most recent 4th edition expansion of his dynamic programming series, or Powell's Approximate DP: Solving the Curse of Dimensionality. Economists are just starting to pick up some of these tools, both directly and indirectly, and I suspect that they will have a growing impact on the literature over the next few years. Some of the analytical background for convergence of these methods is topology and dynamical systems.

A good example of theoretical contribution to this type of literature from economists is Pál and Stachurski (2013), Fitted Value Function Iteration With Probability One Contractions (ungated version here). Peruse that paper and you can see the importance of a good grasp of measure theory. Stachurski's book Economic Dynamics is actually a very nice exposition of dynamic programming from this perspective, building at a pace which works for multiple levels of graduate student/professional (measure theory comes in formally at the end I believe -- I'm still working towards those insights).

Hopefully this answers your question to some degree. I'm afraid that the phrase "post-1960s mathematics" is somewhat ambiguous to me (due to my own lack of knowledge of history of maths literature), so if I've completely missed the mark, my apologies!


Economic effects of a Commons Capital Depository Yanis Varoufakis (Ιωάννης Βαρουφάκης) proposes to fund a universal basic income (UBI) using dividends from a commons capital depository, essentially  channeling a percentage of shares issued at an initial public offering (IPO) into a depository.  He argues that taxing robots is difficult in practice, that funding UBI through income tax is untenable, and taxing capital/wealth is difficult in practice.  Rather, he proposes (see full article for context; emphasis mine):

So how should society be compensated? Taxation is the wrong answer. Corporations pay taxes in exchange for services the state provides them, not for capital injections that must yield dividends. There is thus a strong case that the commons have a right to a share of the capital stock, and associated dividends, reflecting society’s investment in corporations’ capital. And, because it is impossible to calculate the size of state and social capital crystalized in any firm, we can decide how much of its capital stock the public should own only by means of a political mechanism.
A simple policy would be to enact legislation requiring that a percentage of capital stock (shares) from every initial public offering (IPO) be channeled into a Commons Capital Depository, with the associated dividends funding a universal basic dividend (UBD). This UBD should, and can be, entirely independent of welfare payments, unemployment insurance, and so forth, thus ameliorating the concern that it would replace the welfare state, which embodies the concept of reciprocity between waged workers and the unemployed.

Has anything like his proposed commons capital depository ever been tried in practice?  Although he proposes it in the context of UBI, as a form of/alternative for taxation it could be introduced independently.
Is there any historical precedent for such a model?  Are there any economic studies into what the effects would be to shift government revenue from one based on income or capital gains tax, to one such as proposed by Varoufakis?

Has anything like his proposed commons capital depository ever been tried in practice?

Sweden had such experiment, but in a much lower scale than it was originally planned and of what Varoufakis is calling for. This was ultimately due to the opposition of employers/capitalists, and the election of a right-wing government.

The idea came from the trade unions in Sweden, and a proposal was presented in 1975 by the economist Rudolf Meidner. According to this article, its goals were:

to counteract the concentration of ownership which, from a general equality perspective, must be regarded as unacceptable;
to increase wage earners’ influence in economic life
through ownership of capital;
to facilitate the continuation of the solidaristic wage policy by a
mechanism to deal with excess profits (which could also be seen as an element in lessening the concentration of capital)

The fund was to be accumulated by holding a proportion of companies’ profits (originally, 20 per cent before tax), which were to be issued as new shares. This was basically an expansion of equity, to be owned commonly by all workers in the economy. After a few decades, this method would leave most of the (equity-based) companies in the hands of workers. So in effect, this was ultimately a socialisation of the ownership of firms. Quite a bold program.

In the end, a reduced version was introduced in 1984, which lasted only for seven years. The fund was privatised by a right-wing government in 1992.

You can read more about this here and here.

More generally, there are plenty of countries with a so called Sovereign Wealth Fund (SWF). Many of these are financed with natural resources like oil (Norway) and minerals (Chile, copper), either coming from special taxation or royalties, or because firms are owned by the state. These funds tend to be invested in a multitude of assets, not only equity. As such, they are not directly attempting to own capital (as in the case of Sweden). I am not aware of any SWF that yields a citizen's dividend though. Many are used as a buffer in case of an economic crisis, or reinvested in other activities like advanced human capital (e.g. Chile).


Does the government ever pay back its debt from the OMO to the Fed? I'm getting confused about how exactly does money being created by the Fed.

From investopidia.com:

The Fed could initiate open market operations (OMO), where it buys and sells Treasurys to inject or absorb money.

If the Fed holds the Treasurys until maturity doesn't it need to be paid back the par? So how exactly does extra money being created?

Thanks

Yes, it does. Maturing bonds held by the Federal Reserve are repaid by the Treasury, just as they repay any other creditor.

However, the cost to the Treasury of doing this is a policy decision of the Fed and can be effectively zero. It’s critical to understand that the Fed’s excess profits are paid to the Treasury. Last year, they paid $54.9 billion to the Treasury. As a result, the vast majority of the money that the Treasury “pays back” to the Fed flows right back to the Treasury.

To dig a little more into the dynamics, the Fed often decides to reinvest the repaid principal on a bond, so as to not reduce the aggregate money supply (you can think of this as choosing to keep rates lower). When they do this, the actual impact on the Treasury is approximately zero. If they choose not to do so (if they choose to allow borrowing costs to rise), the demand for Treasury debt is reduced accordingly, and the Treasury’s borrowing costs will rise.


Why are demand and supply curves shown as concave up? Why is it that demand and supply curves are typically shown in graphs, including in Econ 101, as concave up (i.e., convex)?

I know what it entails mathematically—that if $p(q_d)$ is a demand curve, $p''(q_d)$ is positive, and if $p(q_s)$ is a supply curve, $p''(q_s)$ is positive, but what’s the intuition for why that’s usually the case?

This is so because usually the marginal costs of firms are increasing in production and many economists think also at increasing rate.

Intuitively you are only willing to provide good Q if the marginal costs are less or equal to price $MC \leq P$. Providing goods to the market at price below marginal cost is irrational and hence inconsistent with basic assumptions behind supply and demand.

The reason why marginal costs are thought of as usually increasing is that the more you use up some factor of production let’s say capital the scarcer it becomes and hence the return to capital increases and hence the MC increases. Moreover, many economists believe that they also do this at increasing rate (That is if you increase the consumption of labor twice the labor costs will increase more than twice) and hence you get convexity. But this being said you can find concave cost functions and also linear cost functions as well which are neither convex or concave in many textbooks as well.


Is there a class of demand functions that deliver equal surplus to consumers and a monopolist? Consider a market with a monopolist firm that has zero marginal cost and faces demand $D(p;\mathbf{a})$, where $\mathbf{a}$ is a vector of parameters and $p$ is the price. The monopolist maximizes profit by solving $$\max_p D(p;\mathbf{a})p,$$
so that the optimal price, $p^*$, satisfies $$D_1(p^*;\mathbf{a})p^*+D(p^*;\mathbf{a})=0.$$

This optimal price, $p^*$, results in consumer surplus $$\text{CS}=\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp,$$
and producer surplus
$$\text{PS}=D(p^*,\mathbf{a})p^*.$$

My question is: is there a family of demand functions, $D(p;\mathbf{a})$, such that $\text{CS}=\text{PS}$ always holds at $p^*$, and if so what does the functional form look like?

We have that

$$D(p^*,\mathbf{a}) = -\frac {d}{dp^*}\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp,$$

$$\Rightarrow \text{PS}(p^*) = -\text{CS}'(p^*)p^* \tag{1}$$

So

$$\text{PS}(p^*)= \text{CS}(p^*) \Rightarrow -\text{CS}'(p^*)p^* = \text{CS}(p^*)$$

or

$$\text{CS}'(p^*) + \frac 1{p^*}\text{CS}(p^*)=0 \tag{2}$$

which is a first-order linear homogeneous differential equation in $p^*$ with variable coefficent. Its solution is

$$\text{CS}(p^*) = B\exp\left\{-\int \frac 1{p^*}dp^*\right\} = B\exp\left\{-\ln p^*\right\}=B\frac 1{p^*},\;\; B>0 \tag{3}$$

So we have that the demand function the OP seeks must satisfy

$$\int_{p^*}^\infty\!D(p;\mathbf{a})\,dp = B\frac 1{p^*} \tag{4}$$

Since it should hold $\forall p^*$ we can consider the derivative w.r.t to $p^*$ on both sides, to obtain

$$D(p^*;\mathbf{a}) = B\frac 1{[p^*]^2} \tag{5}$$

But since, again, it should hold $\forall p^*$, it holds $\forall p$. So

$$\text{PS} = \text{CS} \Rightarrow D(p;\mathbf{a}) = B\frac 1{p^2} \tag{6}$$
Verification of $(6)$ is straightforward.


What arguments are there in favor of national food self-sufficiency? National food self-sufficiency is an idea that appeals to many non-economists.
For example, George W. Bush (2001):

It’s important for our Nation to be able to grow foodstuffs to feed our people. Can you imagine a country that was unable to grow enough food to feed the people? It would be a nation that would be subject to international pressure. It would be a nation at risk. And so when we’re talking about American agriculture, we’re really talking about a national security issue.

I believe that in contrast, most economists view this as a bad idea.
For example, The Economist (2008-05-29), writing during the 2007–08 food crisis:

The argument for self-sufficiency is easiest to counter. Anyone who believes autarky is the route to food security should look at starving North Korea. In world markets trade barriers, not the lack of them, have exacerbated the mess. The commodities that have seen the biggest price spikes are those which tend to be traded least. Only 6% of global rice production, for instance, flows across borders. Unilateral export restrictions, such as those imposed by Vietnam and India, have made matters worse. Global supply is disrupted and domestic farmers discouraged from producing more. The route to deeper, less volatile markets lies through freer trade and fewer distortions. The notion that free trade precludes food security is plainly wrong-headed.

Mohanty (2012):

in the long run, major rice-consuming countries that pursue self-sufficiency may be in danger. If these countries achieved self-sufficiency, import demand for rice would fall. This would push exporters, such as Thailand, Vietnam, and Pakistan, to cut back on their production to reduce exportable surplus and use their lands in planting other profitable crops.
The global rice market, which is relatively small compared with that of other major crops such as wheat, corn (maize), and soybeans, is likely to become even smaller if rice-consuming countries vigorously pursue self-sufficiency. A consequence of a smaller market is greater price volatility and, the smaller the market size, the more the prices have to move in response to any supply and demand shock.
In a year of low production, prices will rise rapidly; in a year of higher production, prices will fall rapidly with greater volatility. This is particularly worrisome since rice production can fluctuate greatly from year to year, at the whim of nature. And, if climate change predictions are realized, extreme weather conditions will be more frequent, leading to regular price spikes in the rice market. ...
A strong global market is essential to achieve global food security for rice. This does not mean that countries should give up rice production and depend on foreign rice. Every country has the basic right to produce enough food for its citizens and this is particularly true for rice, a staple for the world’s poorest of the poor. However, countries might be wiser to try to increase production by improving yield in a sustainable manner rather than pursuing self-sufficiency at any cost.

My question: Have there been any economists who have argued in favor of national food self-sufficiency and what are some of their arguments?

NB. The price of rice rose dramatically during the 2007–08 food crisis (source):

Have there been any economists who have argued in favor of national
  food self-sufficiency and what are some of their arguments?

I am not knowledgeable of literature in that direction, but some of the arguments in the aforementioned excerpts are flawed.

Simply put, self-sufficiency does not preclude free trade. The excerpt of The Economist certainly admits that "The notion that free trade precludes food security is plainly wrong-headed", but it still misses subtle differences between security and self-sufficiency: (1) the latter intrinsically entails more autonomy; and (2) the cost of reliance on international markets might ultimately outweigh the costs of pursuing self-sufficiency.

The undeveloped mention of "starving North Korea" in The Economist improperly conflates the notion of food self-sufficiency and North Korea's policy of generalized isolationism. But, more generally, a country may prioritize food self-sufficiency without necessarily imposing trade barriers.

Note that The Economist article mentions Vietnam and India as examples of economies with export restrictions. But an export restriction is consistent with the purposes of self-sufficiency: It aims at ensuring that the domestic production of the commodity is not depleted by international markets prematurely (that is, prior to satisfying the domestic demand). An increased volatility in international markets as a result of the export restriction becomes irrelevant to the domestic economy if the net effect is that its population is not deprived of the commodity it produces.

Likewise, the issue of "domestic farmers [being] discouraged from producing more" has to do with policy (sub-)optimality regarding the excedents from self-sufficiency. It is not a defect inherent to self-sufficiency.

Mohanty's position fares better in that he rather discourages "pursuing self-sufficiency at any cost". But his argument of climate change predictions is self-defeating because a concentration of supply exacerbates a bottleneck risk --and therewith its potential severity-- of a commodity trade. The global impact of extreme weather conditions is predictably worse in the context of fewer exporters than if sources of supply are geographically more diversified because more economies pursue self-sufficiency.


Why do indebted governments fear deflation? It's widely accepted among people who follow finance that various governments around the world, including the US government, are doing (or their central banks are doing) everything possible to keep their economy out of a deflationary state. (cf. Ben Bernanke's "drop dollars from helicopters" comment.)  The rationale given is that deflation makes your debt more expensive and harder to service.

This rationale doesn't make sense, at least not on the surface.  With inflation, prices in general go up as the value of the currency goes down; with deflation, prices in general go down as the value of the currency goes up.  Therefore, in a deflationary environment, debt service would become more expensive but all the other things on your budget become cheaper, and therefore, as long as debt service comprises less than 50% of the budget, doesn't the country come out ahead?

They don't or shouldn't, fear deflation absolutely. But it's a complicated issue because there are two causes of deflation/inflation. One is changes in the money supply, the other is changes in production.

Changes in production that cause prices to fall are beneficial to everybody. Society becomes materially richer, and this is the great economic success of the last 100 years.

However, prices will also shrink if the money supply falls, since there is less money to buy things with. That in and of itself wouldn't be such a big problem in a market based environment where prices could adjust up and down. However the one thing that can't adjust its price is debt, debtors have made a commitment to  re-payments irrespective of whatever the money supply and deflationary situation is.

It depends where you are in the world right now, as to whether the money supply is expanding or contracting. The situation in Greece is particularly problematic:

This is very similar to the Great Depression pattern in the US which saw a 30% drop in the money supply 1929-1932. Once something like this gets established, it creates self-reinforcing negative feedback loops throughout the economy. For example, part of the reason for the drop is Greeks moving money to banks in other countries that they expect to stay in the Euro. But because of that, Greek banks can't make new loans without violating their regulatory requirements, and that further causes the money supply to shrink.