Why aren't perpetual bonds more common? Most government bonds and corporate bonds have a maturity date when the principal must be repaid.

While the few percents of interest every year is generally not a big problem to pay out, when a bond matures the issuer must pay back all the principal at once which can be a huge expense and the issuer can go default if they don't have all the cash at hand by that time.

If the bond issuer issues a bond without maturity, then they only need to deal with the interest payments, which is a fixed amount of money and as the inflation gradually devalues the money it's become gradually less problem to pay it out.

If the price of the bond decreases or after a good year, the bond issuer may repurchase some of his own bonds from the market on his own discretion to further reduce the interest burden.

From the investor side this is kind of security can be bought from the market and sold when they need the cash just like any other security regardless of the lack of maturity. This way this security can be like a stock with a predetermined dividend.
Since the issuer only need to pay interest, the credit risk is lower, but interest risk is high.

So far it looks like a perpetual bond is just another kind of security with interesting properties and risks.

But most bonds do have maturity, so what are the caveats? What's the reason that most bonds do have maturity and perpetual ones are uncommon?

The most famous perpetual bonds are UK Government Bonds known as consols. They weren't issued to avoid the rollover risk you highlight. Rather, their key benefit was liquidity. They could sell new consuls on the same terms as the old consuls and they have enhanced liquidity because it made the new and old issue more liquid.
They turned out to be a pretty horrible investment in real terms for early investors. Inflation ate away most of their value.
Despite their seemingly infinite life, because they pay out coupons and we discount the future geometrically their interest rate sensitivity behavior is similar to existing long dated bonds.
Unless the bond is callable, you generally can't make money by repurchasing your own bond at market prices. If interest rates drop, making you want to replace old bonds, the price of the old bonds goes up until the yield of old and new bonds are equated. Things are a bit different with changes in credit risk but even in that case it can be tricky.
The product you describe also sounds a lot like preferred stock. The Wikipedia article on perpetuities gives consuls and preferred stock as two examples.


Should free market consumers have the maximum information easily available? As far as I know, free markets rely on well-informed consumers. If this is the case, shouldn't an efficient free market provide consumers as much information as possible about a product so that the consumer can accurately tell the market what they want?
E.g., some people pay more for organic milk. They know which product is organic since it's labelled. What people might not know is whether or not organic milk produces more Co2e to produce per litre. Assuming for the sake of the question that organic milk is more carbon intensive due to reduced yields, how are consumers supposed to know this and weigh up whether they care more about greenhouse gas emissions or animal welfare when buying milk?
Or to take another example. The difference between top-end smartphones is basically the same, but if one product's supply-chain information included "made with child labour and Uigher slaves in concentration camps", people could factor that into their purchasing decision.
I'm not a well-informed consumer. I know I'm not, but there's not a lot I can do about it. (How am I, as a consumer, supposed to find out whether the pair of jeans I'm about to buy is made of cotton from a plantation which is destroying the Aral sea?) But I thought the free market relied on well-informed consumers pushing the market in certain directions. If that's the case, shouldn't products be accompanied with as much information as possible so consumers can make the best choice?
Obviously companies won't advertise the fact they use slave labour voluntarily, so shouldn't an efficient free market be accompanied by the kind of regulation which makes this idea of maximum information possible?

There is an argument to be made that markets work better if there is as little information asymmetry between buyers and sellers as possible. In fact this is so basic insight that you will see it mentioned in pretty much any economics 101 textbook, for example see Mankiw Principles of Economics chapter on asymmetric information.  This is in fact the economic rationale for why modern product packages come with such a cornucopia of information about the content of a product you buy (especially when it comes to food).
However,  in the example above signaling and providing information is essentially costless (companies usually already know what they put into their products - and labels are orientational which is accurate enough - its not that every package of cereal is examined in lab individually). When it is very costly to determine how exactly the goods are produced, the case is less clear cut (it would depend on empirical comparison of cost & benefits). For example,  due to complex supply chains company in US subcontracting to China might not necessarily know about conditions of subcontractor. Requiring all firms by law to have full and complete monitoring of all supply chains might lead to costs that potentially outweigh the benefit of such regulation (again this would require some empirical investigation).
In addition, while it is completely true that:

Obviously companies won't advertise the fact they use slave labour voluntarily

the converse does not hold. Firms have incentive to advertise that they do not use slave labor and so on and hence why so many firms pay money to external auditors to get the privilege of  putting 'fair trade' or 'bio' labels on their products.
This does not mean that government regulation of such labels could not potentially be welfare enhancing. That would require further investigation of particular case, it just means it is not necessary a priori without any further investigation to assume it necessarily requires additional regulation (although my guess is you could probably justify it in non-trivial number of cases after empirical investigation).


Balanced Growth Path (Qualifier Question) $\textbf{Full disclaimer:}$ I am studying for my candidacy exams and this question is from one of the exams last year.

$$max\;\sum_{t=0}^\infty \beta^t \frac{c_t^{1-\gamma}}{1-\gamma}$$Subject to the following constraints:
$$c_t+i_t=y_t$$
$$y_t=A_tk_t^\alpha$$
$$k_{t+1}=(1-\delta)k_t+i_t$$
$$ln(A_{t+1})=ln(A_t)+\mu$$
$$c_t,k_{t+1}\geq 0$$
(b) Compute a balanced growth path in which capital, consumption and output grow at constant rates. On the balanced growth path
(i) What is the growth rate of capital?
(ii) What is the growth rate of consumption?
(iii) What is the growth rate of output?
(iv) What is the ratio of capital to output?
(v) What is the ratio of consumption to output?

$\textbf{My work:}$
Combining the constraints we get:
$$c_t=A_tk_t^\alpha+(1-\delta)k_t-k_{t+1}$$
Taking two terms from the objective function we get:
$$...+\beta^t\bigg(\frac{\big(A_tk_t^\alpha+(1-\delta k_t-k_{t+1}\big)^{1-\gamma}}{1-\gamma}\bigg)+\beta^{t+1}\bigg(\frac{\big(A_{t+1}k_{t+1}^\alpha+(1-\delta k_{t+1}-k_{t+2}\big)^{1-\gamma}}{1-\gamma}\bigg)+...$$
Differentiating w.r.t  $k_{t+1}$ and substituting consumption back in we get:
$$c_t^{-\gamma}=\beta\bigg[\alpha \bigg(\frac{A_{t+1}k_{t+1}^\alpha}{k_{t+1}}\bigg)+(1-\delta)\bigg]c_{t+1}^{-\gamma}$$
$$\implies \frac{c_{t+1}}{c_t}=\bigg[\beta\bigg(\alpha\bigg(\frac{y_{t+1}}{k_{t+1}}\bigg)+(1-\delta)\bigg)\bigg]^{\frac{1}{\gamma}}\qquad(1)$$
I also derived:
$$\frac{k_{t+1}}{k_t}=(1-\delta)+\frac{c_t}{k_t}-\frac{y_t}{k_t}\qquad(2)$$
$$\frac{y_{t+1}}{y_t}=e^\mu\bigg(\frac{k_{t+1}}{k_t}\bigg)^\alpha\qquad(3)$$
My train of thought is the following:
$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t}$ because if $\frac{k_{t+1}}{k_t}>\frac{c_{t+1}}{c_t}$ we would not satisfy the transversality condition because we would have capital in the "last" period which means we cannot be maximizing consumption. If $\frac{k_{t+1}}{k_t}<\frac{c_{t+1}}{c_t}$, we would eventually not be able to maintain constant consumption. Using that and the fact that the ratio of output to capital will be constant over time, equating equations (1) and (2) yields the following messy results:
$$\bigg(\frac{y_t}{k_t}\bigg)^\gamma+\alpha\beta\bigg(\frac{y_t}{k_t}\bigg)=\bigg((1-\delta)+\frac{c_t}{k_t}\bigg)^\gamma+\beta(1-\delta)$$
As far as I can tell, this equation isn't solvable. Does this mean I cannot say that the growth rate of consumption is equal to the growth rate of capital or have I made a mistake somewhere along the way? Also, how would one solve for the growth rates if not this way??

(That one can use the log-difference approximation for the growth rates, can be glimpsed by the fact that while the model apparently is set in discrete time, the log-evolution of technology is expressed using the exponential, which is how we express a constant growth rate in continuous time).

From
$$\frac{y_{t+1}}{y_t}=e^\mu\bigg(\frac{k_{t+1}}{k_t}\bigg)^\alpha\qquad(3)$$

Taking logs you get

$$g_y = \mu + \alpha g_k$$

Where the "$g$"s are growth rates. One the balanced growth path, they have to be equal to some constant, say $g^*$, so you have

$$g^* = \mu + \alpha g^* \implies g^* = \frac {\mu}{1-\alpha}$$

which is the growth rate of the economy on the balanced growth path.

You should also be able (by simmply manipulating the equation with which you start showing your work), to arrive easily at

$$\left (\frac {c}{y}\right)^* = 1- (\delta +g^*)\left (\frac {k}{y}\right)^*$$
So you need only to determine one of the two to obtain the other.

Realize how you can write the left-hand side of $(1)$ while on the balanced growth path, then turn the equation around and solve for $k/y$. Don't expect to get something necessarily "simple-looking".


What is the mix of maturities of outstanding US Treasury bonds called? What is the technical term for the mix of maturities of outstanding US Treasury bonds? Also, is there an "official" source for this information?

I tried searching to find out, but not knowing what the correct term for the maturity mix is, I got no hits.

Weighted Average Maturity of Marketable Debt


Preference ordering relation ≳ is a preference ordering if it is reflexive , transitive and complete.
In Mathematics relations are said to be in a equivalence relation if they are
reflexive, symmetric and transitive.

Can we call preference ordering a equivalence relation?
And if so what is the significance of preference relation being a equivalence relation in further studies of microeconomics.?

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

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

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

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

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

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


MWG Proposition 2.F.1 Is homogeneity of degree zero necessary in proposition 2.F.1?
It seems like the proof does not assume homogeneity of degree zero to establish the proposition.
If this is true, it seems that homogeneity is not required to establish that the Slutsky matrix is negative semidefinite (only required assumptions are differentiability and Walras' law?)

Thank you so much for your help!

In between eq. $(2.F.2)$ and $(2.F.3)$ we read

  ...Walras's law tells us that $w' = p'\cdot x(p',w')$

Assume now that homogeneity of degree zero does not hold. Then we have, $a>0$

$$x(ap,aw) \neq x(p,w)$$

see definition $2.E.1$

Then we can set $p'=ap,w'=aw$ to examine the case $x(p',w') \neq x(p,w)$.

But then Walras' law would imply

$$w' = p'\cdot x(p',w') \implies aw = ap\cdot x(ap,aw) \implies w = p\cdot x(ap,aw)$$

while also, again by Walras' law we have

$$w = p\cdot x(p,w)$$

So under the assumption that Walras' law holds but that we do not have homogeneity of degree zero we end up saying

$$p\cdot x(ap,aw) = p\cdot x(p,w), \,\,\, x(ap,aw) \neq x(p,w)$$

I believe you can contemplate the situation further from here.


How to measure expected price growth rate for oil? How can someone measure the expected annual price growth rate for oil?

Also, where can I find past annual price growth rate for oil values?

Any help would be appreciated.

There is a futures market for crude oil: for example http://www.cmegroup.com/trading/energy/crude-oil/light-sweet-crude.html gives a price now for May 2018 of US$\$49.54$ compared with a cash price now of US$\$47.82$

The futures price being higher than the cash price is described as contango and is why there are excess stockpiles of crude oil.  In the past the comparison has sometimes been reversed, described as backwardation and associated with low stockpiles.  For a graph of this over time, see http://marketrealist.com/2017/02/return-backwardation-mean-crude-oil-prices/ though the relationship is more complicated than described there

For historical oil prices the Federal Reserve Bank of St. Louis has several sets of numbers, such as https://fred.stlouisfed.org/series/WTISPLC going back to 1946


Banks providing loans Banks are providing loans.

While granting a loan two scenarios can take place

The Loan is repaid
The Loan is not-repaid

The benefit from a re-payed loan is small profit. (Let's say 5% in general)
The loss from a not re-payed loan is huge loss. (you lost all your investments)

Are there formulas that can calculate the ratio of loss of (based on pseudo-numbers for example) or other theories related to this issue?

If you have an estimate of the probabilities of each scenario, say with probability $p\in[0,1]$ the loan is repaid (assuming no other scenario exists), then you can calculate the expected value of making the loan of size $L$:

In scenario 1, payoff is $-L+(1+0.05)L=0.05L$ (giving out $L$ and getting back the principal plus interest)
In scenario 2, payoff is $-L$ (giving out $L$ and getting nothing back)

The expected payoff from making the loan would be
\begin{equation}
EV=p(0.05)L+(1-p)(-L)=1.05pL-L.
\end{equation}


Volume of trade covered by regional trade agreements Is anyone aware of statistics about the total volume of trade that happens under regional trade agreements (RTAs)? It's easy to find figures for the number of agreements, but not so easy for their cumulative impact.

What you're looking for is called "utilization rate" and I believe a government agency in most countries should have this information. Here's one example: https://english.bdi.eu/article/news/the-preference-utilization-rate-of-eu-free-trade-agreements/


Why does the United States' national debt matter? This might sound like a naive question, but I mean it quite honestly (please be aware that I am merely a high school student): Why does the United States' government debt matter? To be sure, based on political affiliation there are different degrees of opinion as to how urgent the problem is, but few seem to contest that the ballooning national debt is at least somewhat a concern.

I must confess, I don't quite get it. Given that the United States is so powerful, can't the government just keep borrowing from other countries? (And then just pay off the debt to Country A with more borrowing from Country B, etc., etc.?) Why can't the United States do that forever? Is there really a legitimate danger that other countries will get scared to lend us money (I'm assuming that's the concern about large debt)?

I know other people have asked questions about government debt on this site before, but I haven't been able to find an answer that squarely addresses my present concern, so I thought I'd share this question with you all. Any help is greatly appreciated.

Government debt generally (not just for US) matters (but is not necessary a problem) for country's economy for several reasons:

The most obvious one, generally speaking, debt is not just free money. You have to pay interest rate on debt which costs money that could be otherwise put to other uses. For example according to the US. Treasury the interest costs of US debt in 2019 were almost 600 billion dollars. Now thats not much compared to the size of US economy, but it is not trivial amount as well. Also, this cost would increase with the size of government debt, so even though at current level the debt costs for US are not that large (relative to economy size), if US government debt would continue to increase its debt at rate that outpaces economic growth then eventually this would became a problem.
Having large government debt limits government ability to stimulate economy during recession. There are limits on how much government can borrow as pointed out by @Giskard in his comments. In recession it usually is optimal for government to run deficit to stimulate the economy, but that can be only done if there are people willing to borrow it enough money. Maybe borrowing from other countries could be affected by US soft power (that is question for political scientists) but most of the US debt is own by private people/investors (see here). Generally if people will think that likelihood of US to monetize or default on its debt is high they will be reluctant to lend more money.
Connected to the previous points, if debt to GDP is so high government cant raise enough revenue to pay for interest and nobody is willing to borrow US more, it has to either monetize its debt or default on it. Monetizing debt (creating new US dollars to pay for it) could potentially lead to high inflation, and while little bit of inflation is good for an economy high inflation rates can be disastrous. If US would potentially default on debt it would restrict the US ability borrow in future (however, note that at current level of debt to GDP US is nowhere near a point where this would be a concern. US bonds have still very high rating meaning markets think the probability of US default is incredibly small - in fact US bonds are still often used as an example of risk-free asset).
Governments and businesses are essentially competing for borrowing up peoples savings. Large government borrowing might decrease the amount that private business and individuals can borrow to fund their projects. Even when government debt is sustainable in the sense that debt to GDP ratio is stable, from social welfare perspective having large government debt can be bad, especially if you think that private sector could invest the money more efficiently (See Blanchard et al Macroeconomics, an European perspective for more discussion of this). This does not seem to be huge problem in the present for US because low interest rates indicate excess of savings, but it could be problem in future or at higher levels of debt to GDP.

Note that this list is not necessary exclusive, but I think it provides overview of main points that could cause economists to worry.

Also note, none of the above implies that gov. debt is necessary a bad thing. Depending on situation there is always some optimal level of public debt to GDP ratio (and this optimum in most cases is above 0). However, it is also not sustainable to have debt that is not on a stable path (debt that grows faster than the economy), and even sustainable levels of debt put drain on the economy unless the debt is used on projects that objectively have the highest marginal rate of return which is probably true for some of the gov. spending but its very unlikely to hold for all of it.

Furthermore, I don’t know many economists who would say that current level of debt to GDP in US is huge problem, rather many economists are worried that even when US economy is as strong as it is now, US gov keeps the debt to GDP more or less constant instead of trying to reduce it now so there is more fiscal space for dealing with next crisis. Most of those economists that worry about US debt do so not because of its size now but because of worry that it will increase to more unsustainable levels in the future if US gov. debt to GDP continues to increase.


Is the capital-output ratio in developing countries low or high? For a developing economy, I've seen sources claiming that the capital-output ratio (more commonly known as ICOR) is high (e.g. here and here), while other sources claiming that it's low, like below (the answer is A):

I suppose I can come up with reasons why the ICOR or a developing country might be low: low rate of saving -> little investment on capital -> low returns -> low ICOR. Then again, the ICOR is a ratio, so even if both nominal investment and output are low, there is no guarantee that they're values are therefore low relative to each other, if anything the ICOR can easily be high, as explained in the two sources that claim the ICOR is high for such economies.

The key to the answer is good data on Capital. There is a project (KLEMS), which is computing harmonised (i.e. comparable) information on capital, labour, energy, etc for many countries. At the moment it has information mainly on developed countries, but data for more developing countries are coming up.

For example, this is a calculation of the capital-output ratio for the case of the UK:

For the case of developing countries, here is a figure:

(source here)

It seems that developing countries might have lower capital-output ratios. However, this is ultimately an empirical question which requires a proper analysis. I suggest you use the KLEMS data to make your own, harmonised study.


Derivative of definite integrals - how did MWG arrived at this result? Microeconomics For a wage as a function of profit: $w(\pi)$ and profit $\pi \in [\pi_{min},\pi_{max}]$, the owner of a company sets the minimum wage to satisfy the following condition:
obs $e = \{e_l, e_h\}$ but in this case effort is observable and contractible, and $f(\pi|e)$ is the pdf of $\pi$, thus:
$$\min_{w(\pi)}  \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi$$
subject to manager's participation condition:
$$\int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi-g(e) = \bar{u}$$
after doing a lagrange optimization F.O.C
$$-f(\pi|e) + \gamma v'(w(\pi))f(\pi|e)=0$$
What exactly I do not understand is how the author took the derivative of the integrand and disappeared with the integral signs... why does it work?
edit: Just in case, you can find this situation in MWG page 480-481, but it happens often in microeconomics.

The minimization problem
$$\min_{w(\cdot)}  \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi$$
s.t
$$\int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi-g(e) = \bar{u},$$
is evidently an infinite dimensional optimization problem. The FOC, a la Lagrange, comes from standard consideration for such problems.
To make this more explicit, define the objective and constrain functionals $\Phi$ and $G$ (on an appropriate function space, say the Banach space $C[\pi_{min}, \pi_{max}]$) by
$$
\Phi(w(\cdot)) =  \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi
$$
and
$$
G(w(\cdot)) = \bar{u} - \int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi + g(e).
$$
Then the problem is simply
$$
\min_{w(\cdot)} \Phi(w(\cdot)) \;\; s.t. \;\; G(w(\cdot)) = 0,
$$
which is a standard sort of optimization problem over an infinite dimensional space.
The necessary Lagrangian FOC (as in the finite dimensional setting) would be
$$
D_{w} \Phi + \lambda D_{w} G = 0,
$$
where $\lambda$ is the Lagrange multiplier.
Here, the appropriate notion of the derivative $D_{w}$ is the Frechet derivative.
In this specific case, $\Phi$ is a linear functional. Just as in the finite dimensional setting, the derivative of a linear functional is itself, i.e.
$$
D_{w} \Phi = f,
$$
and, an integration by parts calculation would tell us
$$
D_{w} G = v'(w(\pi))f(\pi|e).
$$
So one arrives at the FOC
$$
-f(\pi|e) + \gamma v'(w(\pi))f(\pi|e)=0.
$$

...this situation ...happens often in microeconomics.

By "this", you mean "solving" such problems by differentiating under the integral sign then setting the integrand equal to identically zero.
Yes, this somewhat loose approach---with various accompanying hand-wavy justifications---often suffices in economic contexts.


How does this demand function satisfy the Weak axiom if it doesnt satisfy the necessary and sufficient conditions? Let $x_k = \frac{w}{\sum{p_l}}$ for $k = 1...L$ be the demand function.

This demand function does satisfy the weak axiom and this can be shown simply.

The slutsky matrix is an $L \times L$  matrix of $0$'s for any (p,w).
In Mas-Colell Microeconomic Theory page 35 and also in this publication - https://www.jstor.org/stable/1911539 , necessary and sufficient condition for the weak axiom to hold are that (i) S(p,w) is Negative semi-definite and (ii) $v^TS(p,w)v < 0 $ whenever $v \neq \alpha p$ for any scalar $\alpha$, $v \neq 0$.

Clearly (i) is satisfied for this demand function and slutsky matrix. However, consider the price vector $[1,1,1,...1]$. The slutsky matrix is still the matrix of zeros. Consider $v=[2,1,1...1]$. $v^TS(p,w)v = 0$ which is not less than 0. Note that $v$ is not proportional to $p$. This appears to violate (ii). Where have I gone wrong?

This is exercise 2.F.17 from Mas-Colell Microeconomic theory.

Your equivalence result is incorrect. The correct results are:

If $x(p,w)$ satisfies Walras' law, homogeneity of degree zero, and the weak axiom, then the Slutsky matrix is negative semidefinite, that is, $v \cdot S(p,w) v \leq 0$ for any vector $v$.
If the Slutsky matrix is negative definite for all vectors $v$ that are not proportional to $p$, that is, if $v \cdot S(p,w) v < 0$ for such vectors $v$, and if $x(p,w)$ satisfies Walras' law and homogeneity of degree zero, then it satisfies the weak axiom.

Therefore, there is no contradiction in the fact that your demand function satisfies the weak axiom while its Slutsky matrix is only negative semidefinite (and not negative definite).


How can non-US banks issue USD loans? I understand, commercial banks are entitled by the Central Bank to "create new money" when they issue a loan and correspondingly "destroy the money" when the loan is paid back (with the constraints of the fractional reserve system). Correct?
If so, I wonder how a Swiss bank, which is entitled by the Swiss National Bank rather than the FED, can issue a new loan in USD. What is the mechanism?
thank you!

In order to understand this its important to understand how actually commercial banks contribute to money creation. It is true that commercial banks are part of money creating process but I think that you misunderstand the way in which they are part of it. When you go to commercial bank to get a loan commercial bank neither domestic or foreign is allowed to just create the money for that loan out of thin air. Only central bank can do that.
Rather the money creation process works in a way that central bank creates base money and in addition also allows commercial banks to borrow from it newly created reserves. Afterwards, the commercial banks help to multiply the base money due to fractional reserve system. For example, if fractional reserve requirement is $10\%$ that means that if a person deposits $\\\$100$ into the commercial bank they are allowed to lend out $\\\$90$ and keep $\\\$10$ as a reserve. Afterwards when the person who borrowed $\\\$90$ deposits them (or spends them and someone else deposits them), bank will again be allowed to keep only $\\\$9$ as reserve and lent out $\\\$81$. In this way eventually up to $\\\$1000$ of new money can be created. This process is reversed once the loan is being paid back. The point is that commercial banks cannot simply create the money out of nothing so their domestic/foreign status does not matter that much.
Foreign commercial banks can make loans in dollars by simply first entering the forex market, buying some amount $X$ of dollars and then lending that amount to its customers. Alternatively foreign bank can also borrow dollars from other bank that has them and then lend them further.  When we talk about dollars in particular foreign banks have even access to federal funds market (see this Fed Cleveland explainer) so they can borrow from there as well.


Does different Base Year in Price Index affected conversion of nominal value to the real value? I do not have any economics background, but currently I have some data sets regarding price index. I have some doubts regarding if my way of conversion is correct?

I have 4 data set. So let it be DS1, DS1 PI, DS2 and DS2 PI.

Note that DS1 PI and DS2 PI are both different variables price index.

DS1 is price index in Base Year 2012, while DS2 is in Base Year 2014.

I am using DS1 to convert the nominal value of attribute 1 to a real value. This adjustment is done to convert all DS1 variables to the current real value at May 2015. The formula is shown below.

$$Adjusted Price 1 =\frac{DS1 Variables}{DS 1 PI} \times DS1 PI_{May 2015}$$

Similarly, I am doing the same thing for DS2.

$$Adjusted Price 2 =\frac{DS2 Variables}{DS 2 PI} \times DS2 PI_{May 2015}$$

Then, I would make a comparison between these 2 adjusted price variables. So my question is since the base year are different, would the adjustment to May 2015 be correct?

Both of the adjusted/real price are now in the same current value and would be comparable. It does not matter if the base year are different in this case?

Sorry if the question is confusing. I would appreciate your help in this matter.

Thanks.

This question might be better suited for a statistics group, but if I understand you correctly you're going to compare two prices from different price indeces and see if there's a difference in the growth? There's really nothing wrong with the reasoning, but you should take the results you get with a grain of salt as you'll be comparing two indeces with different samples.

The different base year is not a problem as long as the variables you're dividing are from the same dataset. When price indices are renewed, the sample is often changed, so this is why you should be aware what you're doing when comparing between different base years.


Any drawback from printing money to pay for national services such as the NHS? Why can't we just print money to build national infrastructure services such as an additional hospital for the NHS? This would create growth, allowing more doctors and nurses to come through but more importantly create additional bed space.

Kind regards

Because increasing money supply will eventually lead to inflation. Consider standard monetary model:

$$MV=PY$$

Where M is the money supply, V velocity of money, P price level and Y real GDP. Velocity of money can be thought of as constant - it’s especially the number of times one euro is used. Moreover, real output can in short run respond to money supply but in the long run money is neutral. Hence when you increase M only thing that changes in the long run is P. You are left with the same output just with higher prices.

Now this simple monetary model is oversimplified- in more modern models it’s expected money supply what matters but the main message stays same.

Also note that in the short run money is not neutral and monetary expansion can boost output but the channel is not really through public investment but through the fact that it helps to put “grease” on the wheels of economy in presence of sticky wages. Hence it might make sometimes sense to just print new money and buy government bonds which will fund also infrastructure or hospitals etc. But most economists believe it has to be done carefully and preferably during recessions.


Mathematical Notations that I Never Understand While Reading Articles or Dissertations I have some problems when I reading mathematical notations. For example:
$$\mathrm{E}_{0}\left\{\sum_{\mathrm{t}=0}^{\infty} \beta^{t}\left[\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)-\int_{0}^{\mathrm{1}} \mathrm{v}\left(\mathrm{h}_{\mathrm{t}}(\mathrm{i}) ; \xi_{\mathrm{t}}\right) \mathrm{di}\right]\right\}$$
This is a optimization equation for representative agent in an economy. This problem here is maximizing this formula.
where
\begin{aligned}
&\mathrm{E}_{0}= \text{expected value in period 0} \\
&\mathrm{C}_{\mathrm{t}}= \text{ index of household consumption of goods supplied in period t}\\ &\mathrm{P}_{\mathrm{t}}= \text{index of prices of goods supplied in period t }\\
&\mathrm{M}_{\mathrm{t}}= \text{nominal money balances in period t }\\
&\mathrm{h}_{\mathrm{t}}(\mathrm{i})= \text{quantity of labour of type i supplied in period t}\\
&\xi_{t}=\text{ vector of aggregate shocks in period t}\\
&\mathrm{u}(.)=\text { utility in period } t\\
&\mathrm{v}(.)=\text { disutility in period } \mathrm{t}\\
&\beta^{\mathrm{t}}=\text { discount factor for period } \mathrm{t}
\end{aligned}
1- First of all, I don't understand the $\xi_{t}$ notation in $\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)$. Does semi colon here mean that this vector notation can affect the equation's behavior exceptionally, but the other variables affect it all the time?.  If there would be a positive or negative shock already, then u and v will be effected through the prices and wages. Why should we use this vector here?
2- Why do we need to take the integral of the disutility function, but don't need to take the integral of the utility function?
3- There are no enough explanations for newbies in articles in mathematical economics, and I can't find a textbook for this topic. I studied Varian's Intermediate Microeconomics, but there is no chapter for this. I looked up books that recommended here but again there is no chapter at all. Is there a book that covers all classical mathematical solutions and explains them in detail for economics?

The usage of ; is not always standardized so it can depend on source. In this case I presume you took the equation from Interest and Prices: Foundations of a Theory of Monetary Policy By Woodford (or some source that used exactly the same notation), as exactly the same expression appears with all variable names being identical on pp 141 (however, in future you should indicate the source).
Based on the usage through the book the author uses the ; to separate variables and shock parameters. Hence, it is just to indicate that the $\xi$ is not a variable but given by the shock parameters of the economy.

Because the arguments of $u$ are already aggregated. $C_t$ is an aggregate consumption index based on the constant elasticity of substitution aggregator:

$$ C_t \equiv \left( \int_0^1 c_t(i)^{(\theta-1)/ \theta}di \right)^{\theta/(\theta-1)} $$
and the price level $P$ is again already aggregate price index:
$$P_t \equiv \left( \int_0^1 p_t(i)^{1-\theta} di \right)^{1/(1-\theta)}$$
So there is already integral over all goods within the utility. Author just written it in this way to 'clean up' the expression. You can always do that by substitution, you could clean it even more by lets say stating that $V(\cdot) = \int_0^1 v\left( h_t(i) ; \xi_{t} \right) di$ and you could then even write it in single equation without any integral at all (this is really more or less an aesthetic/ trying to make your work easily readable issue).

Is there a book that covers all classical mathematical solutions and explains them in detail for economics?

There is no book that covers all mathematical solutions in economics. That would be the longest book ever and you would probably need a truck to move it around. However, there are useful textbooks that cover majority of the most important models in each field.
Intermediate Microeconomics by Varian is undergraduate book so of course it is not on the same level. If you are interested in microeconomics then MWG microeconomic theory or Varian Microeconomic Analysis are the best texts (the former one is quite exhaustive - it is very thick tome).
However, based on the model above I assume you are interested in Macroeconomics. For that Romer Advanced Macroeconomics is a standard graduate handbook. However, even here it would very much depend on exact field you want to study. If you are interested in solely economic growth then Economic Growth by Barro and  Sala-i-Martin is much better than Romer which is more general handbook. If you are interested business cycle then the above mentioned Interest and Prices by Woodford is excellent source. If you like monetary theory then Monetary Theory and Policy by Carl E. Walsh is classic text.


Why Should Expenditure Equal Income in Computable General Equilibrium (CGE) models? I'm reading up on CGE models and one clearing condition that must be met is that Expenditure must equal income.
This seems odd to me because I thought households allocate income to savings and expenditure.

It is because saving is investment expenditure, they must be  equivalent in standard macro models. To explain this forget about CGE models for a second, they are set up in this way because all macroeconomics is set up in this way.
In macroeconomics by definition income = expenditure. This is definitional. For closed economy by definition income (Y):
$$Y \equiv C+ I + G \tag{1}$$
where $C$ is expenditure on consumption (which is what you likely confused with expenditure in general), $I$ is expenditure on investment (public plus private saving), and $G$ is government expenditure (see more details in Blanchard et al Macroeconomics a European Perspective).
Now $I$ will by definition be equal to public and private saving since private saving $S_p$ is by definition income not consumed after taxes: $S_{p} = Y-T-C$. Public saving $S_G$ is by definition any difference between taxes and government spending $S_G = T-G$. Together then all saving must be:
$$S= S_p+S_G = Y-T-C+T-G=Y-C-G$$
Now this must be equal to investment since just by rearranging equation 1 we find that investment is:
$$Y-C-G =I \implies S=I$$
Hence the reason why the CGE imposes restriction that income is equal to expenditure is not that there should be no saving (although there might be simplified models where saving will be 'turned off' to make it easier for students). It is because expenditure and income are macroeconomically equivalent even in presence of saving (which is just investment spending).


The independence of Central Banks How do such digital currencies such as Bitcoin affect the independence of Central Banks?

Thank you

The independence of the central bank means that the bank is independent of the government. A mandate of the central bank is to ensure price stability. However any excess cash the bank prints is the property of the government. The government enjoys having extra income and were it allowed to exercise direct control it may push the bank to print more money. This would undermine the price stability mandate of the bank. Hence in most countries they give the central bank leadership that is independent of the government.

As Bitcoin is not issued by any government that I know of it does not in any way affect this. Perhaps if the government were to bribe the leaders of the bank in Bitcoin...


If price increases and quantity remains the same can we infer that demand is perfectly inelastic? My college tutor told me that demand for college admission at our institute is price inelastic. She says so, because, in the past 20 years, price of college education at our institute has increased, but the number of students hasn't decreased. Does this really mean that college admission at our institute is price inelastic? Or is there a way I can counter that?

We know for a fact that supply is perfectly price inelastic. Each school typically has a quota -- some rough estimate of the number of students it wishes to enroll.

And every year, there are many who are able and willing to pay the full price of the good (a spot in the school) but are nonetheless not admitted.

Example. For decades, a school has consistently offered 10,000 seats to each cohort. Demand though has consistently risen so that price has also risen. However, we cannot infer anything about the slope of D (i.e. whether demand is price inelastic).

That demand is also price inelastic may very well be true, but cannot be inferred simply from the fact that the number of students hasn't decreased.

Another important consideration/complication is price discrimination. Through financial aid, US universities practice what amounts to near perfect price discrimination, making many students pay as much as they can bear. So, while the sticker price has risen greatly, the actual price paid by many students may not have risen as much.


why is Fed creation of fiat money valid? some people suggest that the government prints money to pay its bills and that the increased amount of money in circulation lessens the value of $1.   i hope to get an explanation of why both assertions are false and suggest the following:
the Federal Reserve Bank (Fed) inserts money into the economy thru loans. at least two questions arise

presumably if all loans were paid back, there would be no money in circulation.   but corporations borrow money to build products and others borrow money to buy those products, hence a never ending cycle.  so Fed balance sheet every expected to
be zero (i.e. no outstanding loans)?   and what happens to the money in circulation from defaulted loans?

the "value" of money is presumably based on supply and demand.  this presumably counters the idea that the more money in circulation reduces its value because as the population and economy grow, there is a greater need for money, maintaining its value.   when an economy shrinks, the fed makes fewer loans and as loans are paid back, the amount of money in circulation is reduced.   further, much more money is needed in circulation today than in colonial times (as an example) simply because there is a greater population and need for money (or the economy has grown).

are these concepts correct and do they explain/justify fiat money?  what concepts are missing from a more comprehensive understanding?

If it is good for society to have stable prices then the measure of "valid" money is price stability. The debate over, for example, free banking laws versus central banking customs is one where both teams argue that stable prices are good and each team does not agree on the political-economic customs which would cause the good. So we see that economics involving finance relations (systems of money, credit, and debt) cannot be independent of ethical arguments in the sphere of public debate.
Regarding central bank balance sheet start with the balance sheet identity:

Assets = Liabilities + Equity

The Federal Reserve monitors the sources and uses of Reserve Balances which are liabilities on the Fed balance sheet:
https://www.federalreserve.gov/monetarypolicy/bst_fedsbalancesheet.htm
Reserve Balances and Currency are components of the base money or monetary base which are "valid" means of payment because banks use Reserve Balances to clear interbank payments and society uses Currency to make withdrawals or deposits into banks and for hand-to-hand payment.
The simplified Federal Reserve balance sheet with assets on left hand side of equation:

Treasury Securities + Discount Loans = Reserve Balances + Currency + OLE

where Reserve Balances and Currency are liabilities and where OLE means Other Liabilities and Equity. Fed uses Treasury holdings and Discount Loans as control factors to (1) service the drain of Currency; and (2) provide Reserve Balances to the aggregate bank.
Service the currency drain prior to late 2008 financial crisis:

Treasury Securities = Currency

where if Fed makes its holdings of Treasury Securities approximately equal to the Currency it has provided to banks and nonbanks over the history of Fed operations then this holds Reserve Balances constant to a first case approximation. Before late 2008 there were few excess reserves in the system so Fed had to use gross and fine control over the level of Reserve Balances to enforce control over the federal funds rate.
Fine control over Reserve Balances prior to late 2008 would be accomplished by three mechanisms: (1) Discount window loans; (2) securities repurchase agreements; and (2) reverse repurchase agreements. Since the dual mandate of the Fed is to maintain price stability and employment levels it would not take loans or open market operations (security dealing) down to zero unless political unrest disrupts the nation and puts the Fed out of business as the central bank or monetary authority.

[S]ome people suggest that the government prints money to pay its bills and that the increased amount of money in circulation lessens the value of $1. i hope to get an explanation of why both assertions are false[.]

When people criticize a government for "printing money" this often means the central government engages in deficit spending and the central bank fails to act to curb inflation as a matter of political policy. The inflation is not "caused" by the central bank printing too much money it is "caused" by the failure of the central bank to tighten credit when the bank and nonbank sector drive inflation via too much credit expansion that uses up and makes scarce the slack or excess capacity in the economy.


Is the standard of living of poor & middle class people higher than in the past? I really hope this is not opinion based.  I am just looking for numbers, links, straightforward proof if possible that the lifetime we are in now, this decade, most of us are living better than in the past since the cost of living has increased and so has income inequality.

Are more people on welfare than any time in history for instance?

I think you are looking for data on real median incomes. If this concept is unknown to you, you can read about it on wikipedia.

According to the data the real income of most households decreased after the financial crisis but is still above the 1995 level. This means that most households do live better.

I don't know if the number of people on welfare would be a good indicator of economic prosperity as there have been some changes in the US welfare system recently, drastically reducing the number of recipients.


Is there any specific distribution that is recommended for modelling individual income? I'm a statistician and my colleagues work with income data every now and then, but they usually apply some arbitrary cut-off and go with logistic regression.

I know there's an infinite range of positive support distributions that could work, but I'd like to know if there's something in particular that is supported by economic theory (or perhaps empirical success).

I find it awfully odd to use logistic regression since income is clearly not binary. Sometimes people use logistic to explore if people are in the "top X%" or other sorts of ad-hoc quantile regression. Using logistic seems sub-optimal to me, but perhaps passable. I would prefer quantile regression.

If everyone is employed and you have no unemployed doctors, then you have a wide range of incomes, it makes sense that the distribution would be approximately log normal.  Income distributions have heavy(ish) tails. This is commonly held belief, tied to "Gibrats Law".

Here's an interesting paper on it- carefully note they chose to argue more log normal than. It implicitly confesses of the widely held belief that both consumption and income is log normally distributed.  There are probably more basic papers, but JPE is pretty respected and this has 100+ cites and a good place to dig.

Battistin, Erich, Richard Blundell, and Arthur Lewbel. "Why is consumption more log normal than income? Gibrat’s law revisited." Journal of Political Economy 117.6 (2009): 1140-1154.


Loan money circulation? I have a hypothetical situation and couple of such questions.
Imagine a situation like this:
You need 1000€ a month in order to survive.
There are three loan providers you can borrow the amount of 1000€ for a month with a fee of 1€ a month (you're always repaying money on time).

I borrow money from the first loan provider.
I spend the money.
I borrow money from second to repay the first.
I now owe money only to the second provider.
I borrow money from the third.
I spend the money.
I borrow money from first to repay the second.

And so on and so on, you can see the pattern. It's maybe a stupid question but could loan circulation work as a way of earning monthly money.

This reminds me of the Ponzi Scheme: A Ponzi scheme is a fraudulent investment operation where the operator, an individual or organization, pays returns to its investors from new capital paid to the operators by new investors, rather than from profit earned by the operator. Operators of Ponzi schemes usually entice new investors by offering higher returns than other investments, in the form of short-term returns that are either abnormally high or unusually consistent (source wikipedia with numerous examples).


What are some examples of goods that entrepreneurship-intense? Entrepreneurship is considered to be one of factors of production (the other factors of production are capital, labor and land).
According to Heckscher–Ohlin model, a country that has an abundant supply of a factor of production will have a comparative advantage in goods whose production is intensive in that factor.
I tried to imagine examples of countries with different comparative advantage for each factor:
1.Country A has lots of capital, including human capital. It exports state-of-art semiconductors. It also has lots of programmers, system adminstrators, "white hats" and other IT specialists who export their services by working remotely.
2.Country B has lots of land. It uses said land for farming, consequently exporting products of its farming.
3.Country C has lots of labor, although its workers are poorly educated. People work in sweatshops. It exports clothing.
4.Country D has lots of entrepreneurship. Or in other words, it has lots of talanted and skilled business people. Now I have a problem with this one, I don't know what kind of goods country D could have exported. Can you help?

The term "entrepreneur" is usually taken to mean something more specific than talented and skilled business people.  Wikipedia for example gives several definitions, including characteristics such as (with my emphases):

creation of value;
launching and running a new business;
organize the capital, talent and other resources that turn an invention into
a commercially viable innovation;
willingness to take risks in the name of an idea, spending time and capital on an uncertain venture.

Many who are talented and skilled business people lack some or all of these characteristics.
Entrepreneurship can be applied to many different types of goods.  Taking your countries A, B and C, for example, consider:

Gordon Moore, co-founder of what became Intel;
Parthasaradhi Nara, Indian agricultural entrepreneur;
Nora Bannerman, Ghanaian clothing entrepreneur.

What type of good your country D would export will depend to a large extent on what other factors it possesses in addition to entrepreneurship. As 1muflon1 says there may be opportunities for it to export consulting services, but even that would depend on good transport and/or communications in order to make contact with clients abroad.  But as the above examples illustrate, entrepreneurship is typically applied in order to start and grow businesses using other available factors to develop new products, technologies or markets.


What is the logic behind this sentence in Keynes's General Theory? It seems hard for me to understand the underlying logic of this sentence in Chapter 2 of this book:

  They do not seem to have realised that, unless the supply of labour is a function of real wages alone,
  their supply curve for labor will shift bodily with every movement of prices.

Keynes is talking about the demands of labor is more likely to be a minimum of money wage rather than real wage. Why the supply curve for labor will shift bodily with different prices(prices of wage-goods?) if the supply of labor is a function with other variables(what else variables in this function?)?

I imagine the supply & demand curve intersects on a sector with the x axis being the amount of employment y axis wages, but I'm not sure if this wage should be money wage or real wage or something else.

This sentence says something very simple: If labor supply is a function of many arguments, not just the wage, $L^s = f(w,X,Y,...)$, then the supply curve, which is a line in a two-dimensional space of $(L^s,w)$ keeping everything else constant, will shift as a whole("bodily") if these $X's$ and $Y's$ change.

What could these "$X$'s and $Y$'s" be? First, if the author defined labor supply as a function of nominal wages, then certainly, a change in prices of consumption goods would shift the labor supply curve.

But even if the labor supply is defined as a function of real wages already, then there are other "$X$'s and $Y$'s" that may shift the whole curve, even if the real wage has not changed: for example, direct transfers from the Government (not as a windfall, but as a systematic policy). Since we assume that working is a "bad", people will want to work less for every real wage level, if they get an increased supplementary income through the Government.


Can I add a variable that varies only with time in Least Squares Regression model with a time-fixed effects term? I'm estimating this equation for the trade flows between various countries at time t: ($i$ and $j$ are countries)

The last 3 terms are control variables, one of which is oil prices. Can I add oil prices as a control variable if I already have a time fixed effects term, $ \beta_3 $. Further, can I have country pair fixed effects, $\beta_6$, when I'm also controlling for variables that are fixed for two countries across time (for example, if the two countries were in a colonial relationship in the past, this variable wouldn't change with time and would stay fixed for the two countries for the entire period).

Your model has $\beta_3 * t$, which is a linear time trend, not time dummies. If that's correct, you are controlling for only a linear trend. Because oil prices do not have a perfect linear trend, you can include them.
But I am not sure you really want the linear trend specification instead of time dummies (say, $\beta_{3t}$). For a model with common time effects (year dummies), time series variables are not allowed due to perfect collinearity. Thus, if the time series variables are the key variables, you have a trouble. But if you only want to control for them, you can just omit them because any common time effects will be eliminated by the time dummies.
Country-pair fixed effects are similar. You cannot use time-invariant variables if those fixed effects are included.
One possible solution is to take the correlated random effects approach, but there is the issue of RE $\ne$ FE. You will have to defend your RE model. That's a different issue.


How can a country ensure its national currency is used by its citizens? I am wondering how countries are able to force everyone to use its own currency, instead of some external currency (like the U.S. dollar or Bitcoin or cookies or something else not under government control). If businesses and citizens decide that they do not like the national currency (perhaps it is too turbulent or some other reason), and instead decide that they would like to "barter" a third-party currency for goods and services rather than "buying" those goods and services with the national currency, then the country would lose control of its own currency. This does happen at a small scale, where small businesses may accept a cryptocurrency instead of the local currency.

What different methods can a country employ to ensure that it does not lose control of its currency because its citizens switch to an unofficial alternate currency not controlled by the government?

If this situation does occur, how can a government still collect taxes on the exchange of this alternate money for goods and services? If I wanted to pay someone to mow my lawn in cookies instead of currency, the government could not tax this. If every business and citizen began treating cookies as the main form of payment, the government would lose all tax revenue. How can it stop this from happening?

If the answer involves creating a law that says "10% of all cookies (or bitcoins or something else) must be given to the government", then what prevents that currency from swiftly changing its name? Perhaps cookies become brownies, which do not fall under the law. Or bitcoins become a different cryptocurrency, or just change their name. This would become a game of cat-and-mouse between the laws and the currency. It seems like accepting alternate, nontaxable unofficial currencies, would become economically beneficial for both companies and individuals. Yet established governments in the world do not have this problem. How?

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

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

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

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


Differential equation for first-price auction I want to solve a differential equation of the first-price auction. In particular, from Jonathan Levin's October 2004 lecture notes, we have the following differential equation:
$$b'(s) = (s-b(s))(n-1)\frac{f(s)}{F(s)}$$
Solving this, we should get
$$b(s) = s - \frac{1}{F^{N-1}} \int_\bar{s}^{s_i} F^{N-1}(\tilde{s}) \,{\rm d}\tilde{s}$$
I would like to understand how to get this equation.
I found a November 2011 paper by Timothy P. Hubbard & Harry J. Paarsch discussing this part. On page 6, although it is a regular expression of the differential equation, I should solve the following equation:
$$y=\frac{1}{\mu(x)}\int_{x_0}^x \mu(u)q(u) {\,\rm d}u + k$$
Could you help how to solve this equation to get $b(s)$?

For notational simplicity, let me define the distribution $G(s) = F^{N-1}(s)$ with density $g(s)$. Let $\underline{s} = 0$ (for simplicity).
We have
$$
b'(s)G(s)+b(s)g(s)=s g(s)
$$
Integrating to $x$ gives us
$$
\int_0^x b'(s)G(s)+b(s)g(s) ds = \int_0^x s g(s)ds
$$
Notice that $\frac{\partial }{\partial s}(b(s)G(s)) =  b'(s)G(s)+b(s)g(s)$, so
$$
 b(x)G(x) = \int_0^x s g(s) ds
$$
where we use the boundary condition that $b(0) = 0$.
Now,
$$
 b(x) = \frac{1}{G(x)}\int_0^x s g(s) ds
$$
Finally, apply integration by parts to $\int_0^x s g(s)$ to get
$$
 b(x) = \frac{1}{G(x)}\left[ G(x) x - \int_0^xG(y) dy\right]= x - \int_0^x\frac{G(y)}{G(x)} dy
$$


Stock return predictability. Why do papers continue to use Granger Causality tests There is a long literature on stock return predicabtility.

When I read papers I see lots of authors still using granger causality tests. It is still prominently taught in undergrad and grad classess.

But we do we continue to persist with this when many have shown that they are often not robust to even a little bit of ambiguity. Whether this is persistent variables or in the presense of heteroskedasticity.

Why do people continue to use it?

Why is it so important?

Because the papers which use these methods are not properly refereed. That's why you should read papers published in high impact factor journals.


Confusion regarding debt to GDP ratio While looking at debt to GDP ratio of various countries, I had difficulty with different figures on different websites.
Look at this which shows China's debt to GDP ratio as 43.9%:
http://www.tradingeconomics.com/china/government-debt-to-gdp

Then there are many other links which show China's debt as 250% of GDP.
https://www.theguardian.com/business/2016/jun/16/chinas-debt-is-250-of-gdp-and-could-be-fatal-says-government-expert

What am I missing here?

The answer is in the the first line and title of the articles. Not all debt is government debt. One calculation also includes household and corporate debt.


Repercusions of fabricated GDP (China) China has fabricated its GDP in the past. Which are the real-world repercussions of doing so?

Let me pose an scenario: If China were in a recession (say, -1% for two quarters), but posts 1% for both, which would be the consequences, if the world acted in accord with those figures? Would it impact variables like currency value? Inflation? Interest rates? SHCOMP? Trade surpluses that don't match the published GDP?

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

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

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


Topology on the space of measurable functions The context is as follows:
Suppose we have a 2 period sequential game, with player $i$ in stage $i$, with action set $A_i$.
Give $A_i$ all the nice properties, as compact, separable metric spaces (I'd be happy to even consider $A_i \subset \mathbb{R})$.
A strategy for player 1 is a map $\sigma \in \Delta(A_1)$, and for P2 a measurable function $\eta:A_1 \rightarrow \Delta(A_2)$.
Its clear that when I endow $\Delta(A_1)$ with the weak topology, I can talk about a sequence of strategies having a convergent subsequence.
Is there a topology I can put on $\Sigma=\{\eta:A_1 \rightarrow \Delta(A_2) : \eta \text{ measurable}\}$ that allows me to extract convergent subsequences?

Not really. There are many compact metrizable topologies you can put on this space, but none that relate meaningfully to the structure of the problem.
Let's look first at the case $A_1=[0,1]$ and $A_2=\{0,1\}$. Consider the elevation function $e:A_1\times\Sigma\to\Delta(A_2)=[0,1]$ given by $e(a,\eta)=\eta(a)$. If you want the ultimate action choice of player $2$ to be continuous in player $1$'s action and the strategy choice of player $2$, you need $e$ to be continuous. In particular, $e$ should be measurable when you endow $\Sigma$ with the Borel $\sigma$-algebra. That never works:
It follows from the main result (Theorem D) in Robert Aumann's 1961 paper "Borel structures for function spaces." that there is no Borel $\sigma$-algebra on $\Sigma$ that makes $e$ jointly measurable. For consumers,  I would recommend to read Aumann's 1963 article "On choosing a function at random." instead.
To make this a bit more digestible, let $F$ be the set of measurable functions from the unit interval to itself. Let $e:[0,1]\times F\to [0,1]$ be given by $e(x,f)=f(x)$. If we endow $[0,1]$ with the Borel $\sigma$-algebra and $F$ with any (!) $\sigma$-algebra, $e$ will necessarily be nonmeasurble. Now, by Kuratowski's isomorphism theorem, all uncountable separable and complete metric spaces are isomorphic as measurable spaces. This will include $A_1$ (if uncountable) and $\Delta(A_2)$ (if $A_2$ has at least $2$ elements).


Why is trade openness positively correlated with economic prosperity? Is there any confirmed saying on this problem?

In short: Yes and due to comparative advantage.

The level of GDP growth gained with opening trade does vary by country and time. I'd suggest Wacziarg & Welch (2008). Trade Liberalization and Growth: New Evidence, World Bank Economic Review, vol. 22(2) for a relatively new study on the subject.


Instrument suggestions for FDI I am extremely stuck on trying to think of an instrument for the location of FDI within a country. The instrument must be correlated with foreign firms, but not correlated with domestically-owned firms.

Note: there is no Government policy I can exploit.

Any suggestions, papers, anything at all would be greatly appreciated.

My dependent variable is the log of house price, and my regressors of interest are
1) the log of distance to FDI 2) the log of job creation at the particular firm.

Isolating the impact of FDI on house prices will be difficult, given the presence of domestically-owned investments in close proximity.

Without much information on your regression model it's hard to make a good suggestion. But you may have a look at the Frankel and Romer AER's paper. They construct an instrument for trade, but it could be used for FDI, based on geographic characteristics. I assume that geographic distance between two countries is a good predictor of FDI flows and that ceteris paribus a Canadian firm based in Vancouver has a higher probability to set up an affiliate in Seattle than in Dallas.


Inflation vs GDP growth as factors driving down the US debt-to-GDP ration post-WW2 I'm trying to figure out how much inflation (vs GDP growth) shrank the debt-to-GDP ratio (in the decade or so) post-WW2 in the US. I found one paper that says:

  In 1946, the debt ratio was 108.6
  percent. Inflation reduced this ratio about 40 percent within a decade. [...]

  In 1946, just after the end of World War II, gross Federal debt reached
  121.7 percent of GDP and the share held by the public was 108.6 percent. Over the next 30
  years, debt as a percentage of GDP decreased almost every year, due primarily to an expanding
  economy as well as inflation.

But how much was due the expanding economy and how much due to inflation? (Is that 40% from the abstract just due to inflation?)

Look at the implicit price deflator for GDP. Using the annual series, 1946=11.05, 1956=15.32, so 38.6% increase. (Source: FRED database)

I don’t have time to look at the paper, but that’s possibly what was meant. Not an entirely convincing methodology, but you would have to do something with the deflator as well as nominal GDP to allocate changes.


How can a currency be tied to a commodity like energy? To make a stable (crypto)currency it basically needs to be tied to a thing of value that can not be just flooded onto the market with minimal effort. I think energy is the ideal thing to tie a currency to because basically everything in modern society is dependent on energy or creating or using energy in some form.
But energy can not be easily stored and traded for normal persons so you need some form of stand-in as a currency so i am thinking like 1 "currency coin" is worth 1 kwh. And that to make new coins for the system you have to produce/extract the same amount of energy to the grid. And when that energy is used the "currency coin" is destroyed so there will be a pretty fixed amount of money in the system at any time. And i want it to be decentralized.
But how can this be done? How do you verify that you have supplied the energy to the grid in a secure way? And how can you make sure that the currency value is always tied with the energy value?
I know that bitcoin is kind of tied to energy because you have to use X amount of energy to mine one coin, but this is wasted energy, not energy supplied to a grid to benefit the population.

But energy can not be easily stored and traded for normal persons so you need some form of stand-in as a currency so i am thinking like 1 "currency coin" is worth 1 kwh. And that to make new coins for the system you have to produce/extract the same amount of energy to the grid. And when that energy is used the "currency coin" is destroyed so there will be a pretty fixed amount of money in the system at any time.

This concept does not work with electrical energy.
If we look at “conventional” electrical grids, they ideally would have no resistance, and the grid transmits power (units of energy/second) through the grid without loss, and there is no stored energy. (In practice, the grid has capacitance/inductance, and some energy is temporarily stored.)
The outlined currency scheme would result in a currency unit created and immediately destroyed as used.
The rise of intermittent renewable energy has created a greater need for storage. However, the storage is not in the form of electrical charge in the grid, it is is in some mechanical/chemical store of energy. All storage technologies have inefficiencies, so that energy is lost. There is no one-to-one accounting between input and output energy a currency implies. Furthermore, storage is only useful if it is dissipated during production shortfalls, so the currency backing would disappear each night.
Meanwhile, market prices for intra-day electricity are extremely volatile (since there is no way to carry inventory), so it is not useful as a unit of account.
A currency could be backed by oil reserves - a form of potential energy - but one may note that no oil producer has pegged their currency to oil prices.


Marshall demand for simple CES utility Assume that preferences are given by a utility function is given
$$u(x_1,x_2) = (x_1^\rho + x_2^\rho)^{1/\rho}$$
what then are the Marshall demand given budget constraint
$$p_1x_1 + p_2x_2 \leq I$$

To answer this question I will first generalize slightly the question to deal with the utility function
$$u(x) = \left(\sum_j x_j^\alpha\right)^{1/\alpha}$$
The Marshall demand can be written as
$$x_k^\star(p,I) = \left(\frac{p_k}{\bar p}\right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p} = \frac{p_k^\frac{1}{\alpha - 1} I}{\sum_j p_j^\frac{\alpha}{\alpha-1}},$$
and the value function as
$$V(p,I) := u(x^\star) = \frac{I}{\bar p}$$
where $\bar p := \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$ is the price index as derived here Dixit-Stiglitz Pricing Index.
To find the Marshal demand start from the standard condition that relative prices equal MRS
$$\frac{p_j}{p_k} = \frac{\partial u/\partial x_j}{\partial u/\partial x_k} = \frac{x_j^{\alpha - 1}}{x_k^{\alpha - 1}},$$
in order to get
$$p_k^{\frac{1}{\alpha - 1}} x_j = p_j^{\frac{1}{\alpha - 1}} x_k,$$
which implies that
$$p_k^{\frac{\alpha}{\alpha - 1}} x^\alpha_j = p_j^{\frac{\alpha}{\alpha - 1}} x^\alpha_k,$$
and by summing over $j$ this results in the equation
$$p_k^{\frac{\alpha}{\alpha - 1}} \sum_j x^\alpha_j = x^\alpha_k \sum_j p_j^{\frac{\alpha}{\alpha - 1}} ,$$
which implies that
$$(A)\ \ \ p_k^{\frac{1}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} ,$$
where multiplying with $p_k$ and summing over $k$ results in the equation
$$\sum_k p_k^{\frac{\alpha}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha} = I  \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$
from which I isolate the factor including $x_j$'s to get
$$\left(\sum_j x^\alpha_j \right)^{1/\alpha} = I  \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{\frac{1-\alpha}{\alpha}} = \frac{I}{\bar p},$$
where the LHS is the utility function equal to an expression including only income $I$ and prices which therefore are the value function $V(p,I) = I/\bar p$. Inserting this expression in (A) and isolation $x_k$ gives the Marshall Demand
$$p_k^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} \Leftrightarrow x_k^\star(p,I) = \left(\frac{p_k}{\bar p} \right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}.$$


System of Differential Equations- Asymmetric First-Price Auction I am working on a problem in my Auction Theory textbook regarding a two-player asymmetric first price auction. Assume the bidders are risk neutral. The problem statement is as follows:

    Suppose that bidder $1$'s value $X_{1}$ is distributed according to $F_{1}(x) = \frac{1}{4}(x-1)^{2}$ over $[1, 3]$, and bidder $2$'s value is distributed according to $\text{exp}(\frac{2}{3}x - 2)$ over $[0, 3]$. Show that $\beta_{1}(x) = x - 1$ and $\beta_{2}(x) = \frac{2}{3}x$ constitute equilibrium bidding strategies in a first price auction.

I am trying to work on deriving $\beta_{1}$ and $\beta_{2}$. Unfortunately, my knowledge of differential equations isn't terribly strong. Would someone be able to double check my work and let me know if I have logic errors? I have derived the correct bidding functions, but am not entirely confident my work is sound.

First, suppose the equilibrium bidding functions $\beta_{1} : [1, 3] \to \mathbb{R}_{+}, \beta_{2} : [0, 3] \to \mathbb{R}_{+}$ are strictly increasing and differentiable. Define $g_{1}(x) = \beta_{1}^{-1}(x)$ and $g_{2}(x) = \beta_{2}^{-1}(x)$.

Player $i$ with valuation $v$ can only vary his bid, so he seeks to find the optimal bid given by the optimization problem below.

$$\max_{b} F_{-i}(g_{-i}(b)) \cdot (v - b)$$

We consider the First Order Conditions:

$$F_{-i}(g_{-i}(b)) = \dfrac{f_{-i}(g_{-i}(b))}{\beta_{-i}^{\prime}(g_{-i}(b))} \cdot (v-b)$$

At equilibrium, $v = g_{i}(b)$. Applying this and noting $\dfrac{1}{\beta_{-i}^{\prime}(g_{-i}(b))} = (g_{-i}(b))^{\prime}$, we have:

$$(g_{-i}(b))^{\prime} = \dfrac{F_{-i}(g_{-i}(b))}{f_{-i}(g_{-i}(b))} \cdot \dfrac{1}{g_{i}(b) - b}$$

Plugging in each $F_{i}$, we obtain:

$$g_{2}^{\prime}(b) = \dfrac{3}{2} \cdot \dfrac{1}{g_{1}(b) - b}$$

And:

$$g_{1}^{\prime}(b) = \dfrac{1}{2} \cdot \dfrac{g_{1}(b) - 1}{g_{2}(b) - b}$$

At equilibrium, we have $\beta_{1}(3) = \beta_{2}(3)$. By individual rationality, $\beta_{2}(0) = 0 \implies g_{2}(0) = 0$.

While I could obviously use the problem statement that $\beta_{1}(x) = x - 1$ to conclude that $g_{1}(0) = 1$, I don't know how to justify this boundary condition independently. Does anyone have any insights into this?

Assuming this boundary condition though, I note:

$$g_{2}^{\prime}(0) = \dfrac{3}{2} \cdot \dfrac{1}{1 - 0} = \dfrac{3}{2}$$

From here, I can wave my hand and guess that $g_{2}^{\prime}(b) = \dfrac{3}{2}$, which would imply $g_{2}(b) = \dfrac{3}{2}b$. I'm not sure how to formally derive this though. Would anyone have insights into this?

Once I have $g_{2}(b) = \dfrac{3}{2}b$, I can plug into $g_{1}^{\prime}(b)$ to get:

$$g_{1}^{\prime}(b) = \dfrac{1}{2} \cdot \dfrac{g_{1}(b) - 1}{\dfrac{3}{2}b - b} = \dfrac{g_{1}(b) - 1}{b}$$

Which is a first order linear differential equation, whose solution is:

$g_{1}(b) = b + 1 \implies \beta_{1}(v) = v - 1$.

And we have $\beta_{2}(v) = \dfrac{2}{3}v$.

My work is certainly a little hand-wavy. I would greatly appreciate any help in solidifying the details. Thank you in advance for any help!

I followed Oliv's suggestion, which was quite fruitful. So we have the differential equations:

$$g_{1}^{\prime}(b) = \dfrac{1}{2} \cdot \dfrac{g_{1}(b) - 1}{g_{2}(b) - b}$$

And:

$$g_{2}^{\prime}(b) = \dfrac{3}{2} \cdot \dfrac{1}{g_{1}(b) - b}$$

With the boundary conditions $g_{2}(0) = 0$ and $g_{1}(\overline{b}) = g_{2}(\overline{b}) = 3$, where $\overline{b}$ is the maximum bid.

Now we guess that $g_{1}(b) = \alpha b + \gamma$ and $g_{2}(b) = \delta b + \lambda$. Applying $g_{2}(b) = 0$ yields that $\lambda = 0$.

Next, I substitute $g_{1}(b)$ into $g_{2}^{\prime}(b)$ to obtain:

$$g_{2}^{\prime}(b) = \dfrac{3}{2} \cdot \dfrac{1}{(\alpha - 1)b + \gamma}$$

Integrating $g_{2}^{\prime}(b)$ yields

$$g_{2}(b) = \dfrac{3}{2(\alpha - 1)} ln( (\alpha - 1)b + \gamma)$$

We note there is no constant when integrating, as $g_{2}(b) = \delta b$. We now apply $g_{2}(0) = 0$ again, concluding that $ln( \gamma) = 0$. And so $\gamma = 1$. Thus, $g_{1}(b) = \alpha b + 1$.

We now solve:

$$g_{2}(\overline{b}) = 3 = \dfrac{3}{2(\alpha - 1)} ln( (\alpha - 1)\overline{b} + \gamma)$$

From this and noting that $g_{1}(\overline{b}) = 3 = \alpha \overline{b} + 1$, we obtain:

$$e^{2(\alpha - 1)} = 3 - \overline{b} \implies \overline{b} = 3 - e^{2(\alpha - 1)}$$

Plugging this into $g_{1}(b)$ yields:

$$g_{1}(\overline{b}) = \alpha(3 - e^{2(\alpha - 1)}) + 1 = 3$$

Which implies that the solution $\alpha = 1$. Thus, $\overline{b} = 2$.

So $\delta = \dfrac{3}{2}$.

Thus, $g_{1}(b) = b + 1 \implies \beta_{1}(v) = v - 1$; and $g_{2}(b) = \dfrac{3}{2}b$ implies $\beta_{2}(v) = \dfrac{2}{3}v$ as desired.


How to formulate that a factor may influence a variable or the changes in that variable? I wonder how to formulate how that multiple factors (for instance an announcement of Central Bank A, B and C) may influence a variable. For example, consider the following formula:

$$\Delta I^n_t = f (A_t, B_t, C_t)$$ where $I^n_t$ refers to the interest rate of a security with maturity $n$ at time $t$ and $A_t, B_t, C_t$ refer to announcements of central bank A, B and C on time $t$ respectively.

The above formula indicates that changes in variable $I$ are a function of the announcements of the central banks'. However, how can one introduce this equation? Let me show what my concerns are by the following sentences:

  Changes in $I$ may be due to announcements of Central bank A. However,
  announcements of central bank B and C may also influence $I$

My concern is the last part of the sentence, "influence $I$". Should this indeed be: "influence $I$" or should it be "influence $\Delta I$"? I doubt about this because when the announcements influence $I$ then this should automatically lead to a change in $I$ right?. On the other hand, it is not in line with my equation above. Furthermore, isn't it the case that when some factor influences the changes in some variable, that this variable actually influences the speed of changes in this variable (like a second derivative)?

"may also influence $\Delta I$" seems better to me. Of course for all $s \geq t$ the value of $I_s$ is also influenced by $A_t, B_t, C_t$, but this is an indirect effect. One could say it is fully encoded in $I_{t+1}$.


How do different Indifference Curves relate to each other? We note the following graph

There are three different Indifference Curves on this graph.

  An indifference curve shows a combination of two goods that give a
  consumer equal satisfaction and utility thereby making the consumer
  indifferent.

The goods on the axes are x1 and x2. Assuming they all represent the points at which our economic actor is indifferent to the bundles. How can the person be indifferent to the bundles of goods on each indifference curve, when the lowest indifference curve has less goods in each bundle, than any of the higher ones?

edit: I answered my own question, but would still love it if someone take a look at it, and told me if it makes sense

Each indifference Curve represents a different set of indifferences for different bundles of goods. Assuming the three Indifference curves are labeled I1, I2, I3, with I1 being the bottom one and I3 being the top one.

Take a bundle on I1. The consumer's preference is indifferent to any other bundle that is on I1, but would prefer every other bundle on I2 and I3 to the bundles on I1.


Is there any historical example that a market is shrinking down after competition? I was told that the advertising market as a whole shrank after Google became part of the marketing industry. Apparently, the decrease occurred in offline markets as well as online markets.

Is this a valid historical example (outside of theory) where a large increase in competition in a particular market has decreased the size of that market or was the marketing industry affected differently by Google?

According to The Economist, 2015-09-26, pornography (the "adult" industry) may be one such industry. Due to the entrance of YouTube-like porn aggregators, industry revenues have shrunk dramatically in recent years:

  In America the number of porn studios is now down from over 200 to 20 ... Performers who used to make \$1,500 an hour now get \$500 ... Revenues are well below their peak; how far below is hard to say, as most porn producers are private. Just before the tubes took off, plausible estimates put worldwide industry revenues at $40 billion-50 billion. Mr Thylmann thinks they have fallen by at least three-quarters since then.

Here though we must perhaps be careful about defining "market size". If we define it by such measures as industry revenue and profits, then "market size" has indeed shrunk. But if we define it by say quantity traded (e.g. "number of hours of porn watched"), then probably "market size" is no smaller than before.


Relation $\Delta Unemployment$ and $\Delta GDPgrowth$ I have studied the development of the difference in unemployment and the difference in the growth of real GDP in the Netherlands over the years 1963-2016.

From the picture it seems like a change in unemployment has a positive effect/predicts a change in GDP growth. How is this possible? Has this to do with the fact that unemployment is a lagging variable?

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

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

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


Profit maximizing in monopsonist market The question is as follows: The Happyland Hospital is a monopsonist employer of nurses in the small city of Happyland. The market supply function of nurses is $S(W) = 0.1W - 100$, where $W$ is the nurses weekly wage. What is the hospital's marginal expenditure, ME? If the hospital's marginal benefit of a nurse is $2,000 per week, no matter how many nurses it hires, what is the profit-maximizing number of nurses for the hospital to hire? What will the nurse's wage be? What is the deadweight loss?

And my work so far is as follows:

$Q_s = 0.01W - 100$

$0.01W = Q + 100$

Inverse Supply: $W = 100Q + 10,000$

$ME = W+(dW/dQ)Q = (10,000 + 100Q) + 100Q$

$ME = 10,000 + 200Q$

Profit Maximization - $MB = ME$

$10,000 + 200Q = 2000$

I first derive the Inverse supply curve then use that to derive marginal revenue. Now I'm setting $MB = ME$ as this is the profit maximizing point, but the Q it yields is negative. $(-40)$. Have a feeling this is a problem point. Am I going wrong here?

The question you are showing work out for where $S(w) = 0.01W - 100$ means that you will have no nurses who want to work unless you have a minimum wage of over 10,000 per week, and since the marginal benefit of every nurse is 2,000 per week, this hospital will not be in business. Your original question you typed at the very beginning is making you work with $S(w) = 0.1W - 100$. So let's try approaching that question instead. It also seems you need some help understanding how to get expenditure, so let's go through that too.

First, we find inverse supply.

$$W = 10S + 1000$$

Expenditure is the price of labor times quantity of labor.

$$E = WS = 10S^2 + 1000S$$

The derivative of this is your marginal expenditure.

$$\frac{dE}{dS} = 20S + 1000$$

Set up the profit maximization problem and take the derivative (which we set to zero).

$$\begin{align}
 \pi & = 2000S - (10S^2 + 1000S) \\
 \frac{d\pi}{dS} & = 2000 - 20S - 1000 = 0\\
 & = 1000 = 20S\\
 \implies & \boxed{S^* = 50} \\
\end{align}$$

Then to calculate wage, you just plug in the optimal supply to your inverse supply function. To find dead-weight loss, try drawing out a graph for labor supply and demand. Labor demand should be a straight line across from $W = 2000$. Labor supply was given to you. Notice where the labor supplied equals labor demand. What would $S$ equal there and what wage would be set there? It should be different from the $S^*$ we got. That should give you enough insight to the rest of the question.


Alternatives to Capitalism and Communism Are there any remarkable economists who have envisioned alternative economic system to the ones previously tried? I'm looking for realistic economic system models and their authors.

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

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


How to estimate the labor time required to produce all necessities in a given economy (assuming all goods/services are produced domestically)? I want to estimate how much labor time would have to be spent on producing/providing necessities in an economy if all necessities were produced/provided domestically (without imports). Assuming that we have a clear definition/understanding of what "necessary" means: how would one go about doing so?
My own idea is to use Leontief input-output impact analysis where I first create the Leontief matrix L, check the economic output from the country producing all of its imports domestically x_own_produce, and then calculating x_need by changing the final demand according to the understanding of "necessary". At some point one would also need converting factors to translate from monetary value produced in each industry into labor time (e.g. every dollar of output from a given industry requires 0.5 hour of labor). The way I would change final demand would be to provide the population with a living-wage-budget that they could spend on industries that are considered necessary.
Questions:

Open question: how would you best make such an estimate?

Is my idea a good way to make the estimate? If no: why not?

If my own idea seems legitimate to you: What data would you use? The best data I've found is IO tables from the OECD, but they have relatively few sector categories which worsen the estimate of necessary industries (for example bundling together the production of food products, beverages, and tobacco, where I would consider only the first to be necessary).

In calculating x_own_produce from OECD IO tables would it be correct to do so by setting the import column IMPO: Imports (cross border) to zero for all rows (sectors)?

Open question: how would you best make such an estimate?

Normally, if you would not care about closing the economy, such estimate would be simply done by calculation of real wages per hour and then based on prevailing prices of the necessities you could calculate how many hours would average worker need to work to afford them.
However, the fact that you want to do this for closed economy complicates everything because:
A) Real wage depends on whether economy is open or not.
B) Prices depend on whether economy is open or not.
There are trade models that can help you estimate what prices and wages would be in autarky. For example, you could create some Heckscher-Ohlin or Ricardian model which would be parametrized based on some auxiliary estimates of productivity of various factors at home and abroad (which will have to be done separately) and then simulate what would happen to prices and output based on parameters of the model  (and by extension factor returns (e.g. wages)).
For review of literature on how to estimate factor productivities see (Van Beveren, 2012). Alternatively, you could just do a literature review on countries you are interested in and make separate simulations for range of possible parameters that the model could have (there will always be some theoretical and also empirical estimates that can quite narrow the range of what parameters in the model can be).  For tutorial on how to do numerical simulations of trade model that can then help you determine autarky outcomes you can see this excellent tutorial for GAMS (program specifically built for trade simulations). Even if the tutorial is written for GAMS you can easily use most of what is written there in Python or R.
I can't recommend a specific way of doing the numerical simulation that would be the best because that is very case dependent and this is simply too broad question to go and cover advantages and disadvantages of all possible models you could use here.

Is my idea a good way to make the estimate? If no: why not?

No the idea is not good because of what was mentioned in answer to previous question. Trade generally allows country to consume outside its production possibility frontier (PPF) in autarky and it changes wages and prices.
Generally speaking, countries trade goods that can be produced more efficiently abroad (see Krugman et al. International Economics: Theory and Policy, 11th Edition). This in turn makes prices of those goods lower than they would be absent trade, and in addition it allows country to specialize in what it can do best leading to higher output and consequently real incomes. Of course, there is more nuance to it and trade can also depress wages in some sectors, but the point is that ignoring this would give you wrong and inaccurate results either way.
In your example you are trying to calculate what would happen in autarky using data on prices and wages that exist for open economy, and assuming they would still hold in closed one. This is very unrealistic and will give you wrong results - especially if the economy is small open economy such as let's say Netherlands or South Korea. For very closed economy with minimum amount of international trade such as North Korea (which only has some limited trade with China) it would not matter but I cannot think of any other economy that would not be significantly engaged in international trade. It would be less of an issue for big open economies such as US but only slightly less so.
Given the above the questions in 3 and 4 are moot questions.


How is the rest of the world making the US an interest free loan by holding US currency? In the book "Macroeconomics" by Olivier Blanchard & David R. Johnson, the following appears:

In yet other countries, people who have emigrated to the United
States bring home U.S. dollar bills; or tourists pay some transactions
in dollars, and the bills stay in the country. This is, for example,
the case for Mexico or Thailand.
The fact that foreigners hold such a high proportion of
the dollar bills in circulation has two main macroeconomic
implications. First, the rest of the world, by being willing
to hold U.S. currency, is making in effect an interest-free
loan to the United States of $500 billion

By holding US bills of 500 billion, the rest of the world is making the US an interest free loan of $500 billion---could you please explain me how?

Think about what a loan is in terms of purchasing power.  You give a bank your signature on a loan contract and they give you something that will allow you to buy real stuff (houses, cars, whatever).  You enjoy this real stuff for some time. Then at the end you have to give the bank something that will allow them to buy real stuff, only more of it.  I'm calling it "something" instead of "dollars" for a reason I will explain below.

In the foreign exchange case, the US gives other countries pieces of paper, which are essentially IOUs.  They give the US either real stuff, which is then enjoyed, or they give the US their currency, which is immediately spent on real stuff.  The US doesn't hoard their currency.  Time passes and at the end they give back the IOU's in exchange for some real stuff.  Only instead of giving back more real stuff than was originally given, the US gives back less (because of inflation).

I called dollars "something" in the loan example and "IOUs" in the macro example because dollars take on a different role in each case.  In the loan, the dollars represent real purchases that are made immediately. In the macro example, the dollars take the place of the loan contract.  They don't buy any real stuff until the end of the transaction.

You can think of it another way by ignoring the dollars.  Foreigners give the US electronics and clothes and stuff in exchange for nothing.  After a while the US gives them back food and machines or whatever, only what the US gives back is worth less than the electronics and clothes it was given.  That's a zero interest (or negative interest) loan.

By the way, it's not just foreigners.  Whoever holds currency has given up something valuable (time, materials, land, etc.) in exchange for paper.  As long as that paper is under your mattress, you are not consuming and the paper is becoming worth less than it was.  You've made a negative interest loan.


What are the determinants of the natural rate of interest and how market rate converges to it? I'd like to know how is determined the natural/neutral rate of interest. Moreover, how the market rate can converge to it ? If you have some references it would be great !

What determines interest rates?
Let's try to understand the emergence of interest rates from first principles. We'll imagine an island inhabited only by Robinson Crusoe and Friday. For sustenance, they pick coconuts - the only food source available to them. Each coconut can either be consumed immediately or saved up for future use.
One day, Crusoe does not manage to pick any coconuts and having none saved up asks frugal Friday to loan him one from his stockpile. In return, Crusoe promises to give Friday more than one coconut at some later time. Let's consider what factors would affect how many future coconuts Crusoe must promise Friday in order to persuade him to issue the loan.

If Crusoe is hungry right now, he will have a high time preference (=low patience). Therefore he will be willing to promise a large number of future coconuts for one present coconut.
If Friday has a large number of coconuts saved up, say 100, he will be more willing to part with one of them for a modest fee. If he only has 10 saved up, he will demand more future coconuts if he is to loan out 10% of his savings. How many coconuts Friday has saved up determines the marginal utility of 1 coconut and thus how willing he is to loan it out.
If Friday thinks there is a high risk that Crusoe will struggle to pick enough coconuts in the future to pay back his loan, he will be less willing to issue a loan without being compensated for it.

If more inhabitants are added to the island, we can imagine a market for "coconut loans". Lenders will compete to offer customers an interest rate that correctly balances time preference, risks, etc. Those who offer too high an interest rate will get few customers, while those who offer one that is too low risk going bust.
Note that there is no such thing as "the interest rate". There are only particular interest rates on particular loans made between particular people. Like many other things, talking about the "natural" interest rate and how lenders converge to it can sometimes be a helpful abstraction, but the true, underlying mechanisms should always be kept in mind.
In reality, the picture is more complicated since it is possible for central banks to artificially affect it in a way that is not constricted by supply and demand. I can recommend this article for a discussion of the subject.


Why do central banks currently favor cheap money policies? Currently, central banks around the world are setting very low interest rates.

According to Taylor rules, which most central banks can be shown to roughly follow, this implies that they believe that the natural level of output is greater than the current level of output and/or current inflation is below its target. Looking at real GDP per capita charts (e.g., US, Eurozone, World), I could not see that we are far below the medium-run trend in real GDP. Therefore, I am wondering what current factors drive central banks to implement low interest rates?

Most central banks' primary goal is maintaining a moderate inflation (around 2 %). In some cases, they also have a secondary and lesser mandate to care for employment numbers.

Since inflation is not taking off but persists at a low level, the central banking system has flooded the markets with money. The thought is that the money will "trickle down" from banks and institutions with easy access to money. However, the effects on the real economy have been somewhat limited.

The central banks' idea is that when capital is cheap to come by, companies will invest and thus hire more people to the workforce, who will in turn consume more. This isn't really happening, which is likely due to demographic and technological factors. After employees were released during the 2008 financial crisis, companies haven't really started to employ people again in the same magnitude. To a higher degree, production processes are now outsourced or automated. Companies who are hiring, typically are looking for people with higher education or specialised skills. Most unemployed people available to the labor market do not meet these requirements. So we have deflationary pressures in the world economy that are keeping inflation down.

If central banks really wanted to get inflation going, it would probably be more effective to just hand out money directly to the consumers' bank accounts, than having money pass through intermediaries. The exception to this is that we have seen pocket bubbles in real estate and some other asset classes. Despite interest rates being record-low, people who already have some wealth aren't really consuming much more, neither are they saving funds in their bank accounts, but have instead tried to find asset investment opportunities to increase wealth. This is creating lending and speculation bubbles in real estate and an upswing in stock and bond prices.

Some economists are also starting to question the objective of pursuing inflation and if it really should be the paramount objective of central banks, when the problems created by massive asset bubbles could become so much worse.