Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative The problem:

Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$-Brownian motion.
Let $X = \{X_t\}_{t \in [0,T]}$ be a stochastic process where $X_t = W_t + \sin t$, and let $\mathbb Q$ be an equivalent probability measure s.t. $X$ is standard $\mathbb Q$-Brownian motion.
Give $\frac{d \mathbb Q}{d \mathbb P}$.

Girsanov Theorem:

Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is the standard $\mathbb P$-Brownian motion.
Let the Girsanov kernel $\{\theta_t\}_{t \in [0,T]}$ be a $\mathscr F_t$-adapted stochastic process s.t. $\int_0^T \theta_s^2 ds < \infty$ a.s. and $\{L_t\}_{t \in [0,T]}$ is a $( \mathscr F_t , \mathbb P)$ martingale where
$$L_t := \exp(-\int_0^t \theta_s dW_s - \frac 1 2 \int_0^t \theta_s^2 ds)$$
Let $\mathbb Q$ be the probability measure defined by
$$Q(A) = \int_A L_T dP \ \forall A \in \ \mathscr F$$
or $$L_T = \frac{d \mathbb Q}{d \mathbb P}$$
Then $\{W_t^Q\}_{t \in [0,T]}$ defined by
$$W_t^Q := W_t + \int_0^t \theta_s ds$$
is standard $\mathbb Q$-Brownian motion.

The solution given:
$$X_t = W_t + \int_0^t \cos s ds$$
Let $\theta_t = \cos t$:

It is $\mathscr F_t$-adapted

$\int_0^T \theta_s^2 ds < \infty$ a.s.

$E[\exp(\frac 1 2 \int_0^T \theta_t^2 dt)] < \infty$

Then $\{L_t\}_{t \in [0,T]}$ is a $( \mathscr F_t , \mathbb P)$ martingale, by Novikov's condition, where
$$L_t := \exp(-\int_0^t \cos s dW_s - \frac 1 2 \int_0^t \cos^2 s ds)$$
Thus, by Girsanov's Theorem, we have
$$\frac{d\mathbb Q}{d\mathbb P} = L_T...?$$

How exactly does that last line follow?
What I find strange is that the Girsanov Theorem defines $\mathbb Q$ and then concludes $X_t$ is standard $\mathbb Q$-Brownian motion while the problem says there is some $\mathbb Q$ s.t. $X_t$ is standard $\mathbb Q$-Brownian motion and then asks about $\frac{d \mathbb Q}{d \mathbb P}$. Is the problem maybe stated wrong?
To say that $L_T$ is indeed the required density $\frac{d \mathbb Q}{d \mathbb P}$, I think we need to use the converse of the Girsanov Theorem), or maybe the problem should instead give us $\frac{d \mathbb Q}{d \mathbb P}$ and then ask us to show that $L_T = \frac{d \mathbb Q}{d \mathbb P}$ possibly showing that $E[\frac{d \mathbb Q}{d \mathbb P} | \mathscr F_t] = L_t$ or some other route.

I tried something slightly different:
I define $\hat{\mathbb P}$ s.t.
$$L_T = \frac{d\hat{\mathbb P}}{d\mathbb P}$$
or
$$\hat{\mathbb P} = \int_A L_T d\mathbb P$$
It follows by Girsanov Theorem that $X_t$ is standard $\hat{\mathbb P}$-Brownian motion. Since we are given that there is some $\mathbb Q$ equivalent to $\mathbb P$ s.t. $X_t$ is also standard $\mathbb Q$-Brownian motion, it follows by the uniqueness of the Radon-Nikodym derivative that
$$\frac{d\hat{\mathbb P}}{d\mathbb P} = \frac{d\mathbb Q}{d\mathbb P}$$
$\therefore, \frac{d\mathbb Q}{d\mathbb P}$ is given by $L_T$.
Is that right? I think I'm missing a step somewhere.
So, is that indeed what the solution given is meant to be but just omitted pointing out uniqueness of the Radon-Nikodym derivative, if such justification is right?

Edit based on this: Even if Radon-Nikodym derivative is unique, $\mathbb Q$ may not be unique? If so, is it then that $\hat{\mathbb P}$ is merely a candidate for one of many possible $\mathbb Q$'s?
I think we conclude $\hat{\mathbb P} = \mathbb Q$ based on $X_t$ being standard Brownian motion under both measures. Is there a proposition for that? Uniqueness of Brownian motion measure or something?

As you say (how did $\sin t$ in the initial problem become $\cos t$?), $\mathbb{Q}$ is a measure under which $W_t$ becomes $\tilde{W}_t - \sin t$, where $\tilde{W}_t$ is a $\mathbb{Q}$-Brownian motion. So Girsanov's theorem would say

$$
L_t = E[\frac{ d \mathbb{Q} }{ d \mathbb{P} }|\mathcal{F}_t] = e^{ - \int_0^t \sin s dW_s - \frac{1}{2} \int_0^t \sin^2 s ds}
$$

is one such measure.

As for uniqueness, if we take $\Omega$ to be the canonical path space $C[0, T]$ with $\mathcal{F}$ generated by the cylinder sets. Then the Wiener measure would be unique, simply by the fact that the Brownian law says any two such measures must agree on the cylinder sets and therefore all of $\mathcal{F}$.

In general, the Girsanov $\mathbb{Q}$ is unique up to the Brownian filtration $\mathcal{F}_{W} \subset \mathcal{F}$, since it is constructed $W$-pathwise.


If you short an ETF, who pays its MER? I don't understand how u/Cruian's comment answers the original questions in that Reddit post.

The expense ratios are taken from the fund's holdings directly, you never see the change in your account directly.

So does the borrower of the ETF pay the [Management Expense Ratio ("MER")? The lender?

If the borrower must pay the lender the dividends, is the borrower credited the MER? Or is the borrower debited the MER?

As the answer says, the management fee is directly taken from the fund’s assets. The only way that anyone “pays” the fees is via lower returns. There is no cash flow to pay the fee.


Why are some goods without close substitutes not sold in some countries? Especially in consumer electronics, a large number of goods are offered only in some countries. Usually this holds only for products which have some close substitutes available (e.g., screens, where a particularly cheap models are often only sold in the U.S. but not in Europe). Such cases are not too hard to explain via price discrimination.

However, I recently found out that there exist products which do not have close substitutes such as Sony's E-Ink readers but are nonetheless only available in some countries. (Production limitations were my first thought, but the example product is already several years old.) Given that the online retailer market is very efficient and that consumer electronics have large development costs I could not come up with a reason why consumer electronics producers would willingly restrict their own market size.

Since I do not believe that producers make such decisions without good reasons, I would like to know what these reasons are. Note that the example product is not the only case, similar cases can be found earlier in the cell phone and PC markets.

One way to look at this is through the idea of "fixed costs". It only makes sense to pay the fixed cost if the market is big enough.

There is very likely a large a fixed cost of entering a market. You have to establish a network of retailers or at least distribution channels, a way for the customers to return the defective units, partners that will also supply the covers, the software, the adapters, etc. Often you have to register the devices with the local regulators, they have to approve them for sale in that country, etc.

From the marketing perspective there are also fixed costs. You make a single advertisement campaign per country, and you use it everywhere in that country. I suppose there is a minimum word of mouth effect that needs to take place for any marketing to be available, so I suppose a small marketing campaign is unlikely to be efficient.

Also, sometimes products are sold or presented basically as flagship products, or loss-leaders, to call attention to the technological prowess of a firm, but not because they are all profitable. So, if a brand does not face a lot of competition in a country, it might not introduce these products as selling them, might actually represent a marketing expenditure.


Meaning of UK Employment Change Index - 3 Months Average I am trying to understand the meaning of an economic index which is measured in the following way:

The value for month n is determined by calculating the average of the last 3 months (n, n-1, n-2) and subtracting the average of the previous (non-overlapping) 3 months (n-3, n-4, n-5).

I wonder if anyone has an idea as to how to isolate a certain month (that data is not published) or how to better "understand" the meaning of the value of a specific month - the economic meaning (difference between one month and another) is not that clear to me.

Thanks, Tom

Let's say the raw series is $x_1, x_2, \ldots$. The three-month moving average is given by $y_n = \frac 1 3 \cdot (x_{n-2}+x_{n-1}+x_n)$. What you have, though is $z_n = y_n - y_{n-1} = \frac 1 3 \cdot(x_n - x_{n-3})$.

Without knowing any of the $x_n$'s, you can't back out $x_n$'s from $z_n$'s. Knowing only $x_1$, however, would give you $x_1, x_4, x_7, \ldots$. It's likely that you would know this, since the series we're interested in is an index, which usually gives the base year.

In all, you need to know at least $x_1, x_2, x_3$ (or, to be more precise, at least one $x_i$ where $\mod(i, 3) = j$ for $j \in \{1, 2, 3\}$ to be able to back out the entire series of $x_n$'s.

Usually, people are interested in three-month moving average because the value for each month is highly volatile and can't be used with much precision.


Why can't a government print money to pay its debt? I have written an explanation to one of the famous questions "why can't government print money to pay its debt".

Is my explanation correct?

Is this the similar situation which happens in real life?

Is there really a foreign exchange bank?

Do companies like Verizon really buy other countries' currencies?

In Country X, President thinks we have too much debt to be paid to USA. Let us print money and will pay the debt off. Country X uses currency Y.
After printing money, since X has to pay to USA in dollars, X goes to foreign exchange bank and try to catch some "fool". Suppose a telecom operator in USA say Verizon wants to expand business in  X, wanted to buy Y since X may need that money to construct office, hire people, bribe politicians etc. So Verizon buys it from X in return for USD. Now X pays its debt off to USA in USD and sleeps happily.
Now Verizon expands business in X, effectively returns the money they got it from X to the people of X, by paying for construction, employees, etc.
So people of X have more money but the production of X remained same. This is a classic case of inflation. So countries tend not to print money to pay off debt!

Your get the basic intuition, but real-life situation is a  bit different.

First, I didn't really understand what you meant by "try to catch some 'fool'", but what you are calling foreign exchange bank is actually called the Foreign Exchange (market), and consist of buyers and sellers that are basically ready to buy and sell any currencies (almost all currencies in fact) provided that the price aligns with their expectations. Classic laws of market apply to this market, and if X's government (its central bank to be more precise) prints a lot of money, money supply increases. The result is a decrease in the price of money relative to other money, which is the exchange rate.

Money may be hard to understand, but it is actually a right to future cash flow of a given country. It is a sort of perpetual debt with zero coupon issued by a given country. So, if the country issues more money, it means that it issues more debt, so as a dilution mechanism it yields less cash flow to each unit of money, and since the price of money (as the price of any debt) is the expectation of future cash flow, the price will decrease.

Then, this will indeed result in more inflation. I expose two ways of seeing that. First, you can think about the price of goods as the price of goods relative to the price of money. Indeed, you're going to pay $x$ for a particular good because the seller knows that it will receive $x$ and that $x$ will give him the right to a certain amount of GDP (cash flow of country). But if the amount of GDP that the money represents is lower, he will ask for a higher price. Second, traditional monetary research has derived, a long time ago, the following formula which represents what I explained in my first point, but more formally. The following is always true, it is an accounting relation :
\begin{equation}
M\cdot V = P\cdot Q
\end{equation}

$M$ is the amount of money, $V$ is the speed of money, $P$ is the price level and $Q$ is the output or quantity of goods exchanged.

Without entering into too much details on this equation (there are extended information on Internet), you can see that increasing $M$, holding $V$ and $Q$ constant, results in inflation.

To come back to your question, countries don't (only) print money to repay their debt, because they are afraid of inflation. There is a big debate on inflation (You can read it on Internet), but loosely speaking inflation seems hard to contain. So to avoid hyper-inflation situation, they try to maintain inflation (in general below 2%).

Your story does not need another company investing in  X because, as you said,  people will end-up with more money, but X's production  will effectively increase. But it is true that companies buy currencies for investment purposes in their own country. Even local company may end-up buying their own currency, if for instance they sell product in USA (exportations), and sell the USD they receive against buying their own currency. This is why import and export (worldwide) are good indicators of exchange rate fluctuations.

Printing money to pay debt is a complicated question, and I only tried to sketch some insight here. For more information, read more on money creation and inflation (either in the academic literature or in economic textbook for instance).


Is NIIP a good measure of a country economical development? In another question I asked someone pointed me to an answer where NIIP (Net International Investment Position) was mentioned as a measure of economical development, which I already heard about it and it makes sense since it takes into account not only the actives but the pasives of a country

But then you see this list, and you find out that first world countries like United States, Australia, Sweeden or Spain are in big red. What does this mean? Does it mean that NIIP is a poor index to measure economical development or wealth? Or does it mean those countries are in a serious bad situation which could lead them to be in a crisis in the future?

For common definitions of the word development the NIIP would be a very bad measure. The NIIP is a measure of net foreign debt. I don't see any clear first order link between foreign debt or assets and economic development.

A popular measure of development is the HDI, which is constructed from data on live expectancy, adult literacy rates and GDP.


A question about savings and investments Hey all I am new to macroeconomics and I'm studying right now. I see in my notes that savings $=$ investments, $S=I$.
I've searched a lot online to see why this is true and I've found a lot of answer but all of them include concepts that I don't yet know, or complicated mathematics.

Is there an intuitive explanation you can give me, maybe with an example, that doesn't involve mathematics or complicated examples about interest rates etc?

The way I see it is, the total amount of money that can be invested in a country can't be more than the savings of a country. If me and someone else were the only 2 people living in country $X$ then if I saved $500$ dollars today, then $500$ dollars is the total amount of money that can be invested tomorrow for example. But I don't know if this a right way to see it.

Thank you in advance.

It is a statement about national accounts that has to hold (an accounting identity). Unless you understand all the terms in the national accounts, the accounting identity may not make sense.

One non-obvious property of the $S=I$ accounting identity is that increases in inventories are considered investment. That is, it is not just fixed investment (machinery, buildings) which people normally associate with “investment.”

Take the simple example of a firm that sells goods with a zero profit margin. Its workers are its customers. It pays its workers \$100, and they save \$10. This gives it sales of \$90, and a build up of inventory of \$10. The savings of the workers did not go to fixed investment, it resulted in a possibly undesired build up of unsold goods at the firm.

One thing to note that we are discussing flows within a single accounting period; in my example, the \$10 build up of inventories happens in the same accounting period as the savings (the same day, for example). As a result, when you write in the question that the \$500 savings today can be used for investment tomorrow is incorrect if we our accounting period is one day at a time - the savings today has to match investment today. (If we had a longer accounting period, such as quarterly, today and tomorrow could end up in the same accounting period.)

As a result, in order to understand the identity, you need to understand the definitions of the various components of the national accounts. If you added an example of a discussion that you had difficulties with to your question, someone might be able to give you more guidance.

My trick to looking at this is to note that the accounting identity holds for any accounting period, including extremely short ones. In particular, any group of transactions satisfies the identity. I then try to see how the low level transactions would end up if they were the only transactions in the national accounts (like the example above).


If a utility function is quasi-concave, can we say that the IC curve associated with it is convex? Let's say we have an utility function, $ U(x,y) = \sqrt{x \cdot y} $. The indifference curve associated with this is convex, while the function itself is quasi concave (because it satisfies $ f_{xx} f_x^2 - 2 f_{12} f_1 f_2 + f_{yy} f_y^2 $).

So, can we say that an utillity function is quasi-concave if the indifference curve is convex (and vice-versa)?

In your example the IC curve is not convex in the usual meaning of the word convex when applied to sets. What you probably mean is that the IC curve implicitly defines a convex function $f$ where $f(x) = y$.

If this is indeed what you mean then as long as you also assume that $U$ represents monotone preferences, then the quasi-concavity of $U$ will imply the 'convexity' of the IC curve and vice-versa. In this case the upper contour set of any level of $U$ will be a convex set, the lower border of which is the 'convex' IC you mention.


How can the GDP's growth rate be positive, when the GDP decreases? I have question about following charts in Mozambique. Here is GDP of Mozambique in 2010-2016 - https://data.worldbank.org/indicator/NY.GDP.MKTP.CD?end=2019&locations=MZ&start=2008. You can clearly see that GDP grows from 11 billion to 17.8 billion and then lowers in 2014-2016 from that number to 12 billion dollars. However, the GDP growth rate remains positive (between 6-7 % - https://data.worldbank.org/indicator/NY.GDP.MKTP.KD.ZG?end=2019&locations=MZ&most_recent_value_desc=false&start=2005). How is it possible, shouldn't the "GDP growth" be negative?
I understand the growth rate isn't as easy as year-2-year change. Perhaps there is inflation (that was pretty low in Mozambique, around 2%), maybe exchange rate plays a role as well?
Thank you a lot for answer!

Your first link is "GDP (current US$)".
Your second link "GDP growth (annual %)" is "based on constant local currency" (from "Details").
For figures that correctly correspond to your second link, see instead this third link: "GDP (constant LCU)" (LCU = local currency unit).
The discrepancy between the first and third links is explained by the great fall in the Mozambican metical (against the USD) in 2015 and 2016:


Economics term for unfavorable goods What are called goods, whose maintenance expenses are higher than initial price tag. There is a specific term for that, but I can't find it.

You might be thinking about Demerit goods. This type of goods have negative externality (cost) that are usually ignored by or unknown to the consumer.
As a result, initial cost of the good is relatively low, when compared to the long term cost for the consumer.

Simple example of a demerit goods are cigarettes, since they have to fulfil two criteria:

be harmful, unhealthy to the individual consumer: smoking is accepted as an unhealthy habit.
have negative externalities, cost to third parties: secondhand smoke.

Finally, it is important to notice that classification of demerit goods is normative contrary to usual positive economic approach when we classify goods based on their price or income elasticity e.g. ordinal, inferior, luxury.


Are there any major commodity markets with a public order book? Most major exchanges seem to have secret order books.

For example, on the NYSE only the specialists know the order books.

On the NASDAQ only the market maker can see the order book.

On Globex, the full order book (Level 6) is kept secret.

Are there any major exchanges with a public order book?

Example of a public order book:

https://www.bitstamp.net/market/order_book/

Major trading exchanges have order books that anyone can see, if they're willing to pay for the service. So in that sense, they are available to the public. For example, anyone can subscribe via a broker to the London Stock Exchange Level 2 data, which shows them the live order books for listed instruments.

Very few have "secret" order books. I think you mean dark pools; they do exist too, and wikipedia has a list.


International date line What if international date line being shifted to Atlas Ocean? Most of trade partners will have different date, States and all America continents will enter day first etc. So what can be coincidence of this? I am asking this because I have read that samoa was shifted to have better trade partnership with Australia.

The consequence would be confusion, if someone is further east of you it's natural to think they're ahead of time (relative to you).  Especially if you're used to the ubiquitous maps of the globe (with Europe/Africa in the middle, the Americas to the left and Japan/Oceania on the right).  Confusion then might arise in some economical contexts that are highly standardised.

For example, spot transactions on the forex markets are delivered (cleared) in 2 days.  If someone behind the date line trades on your behalf you might have to wait 3 days, or just one day for the funds to clear, depending on what side you're on.  Also in your example, think of recurring events (like quarterly reports, dividends, etc.), you might still be on the day before they're due but time-wise (just the hours on the clock) you're ahead.  Or if you're west of the date-line and waiting for events from someone east thereof you have to wait another day.

If the explanations above are confusing then that's exactly the point I'm making.  So, my opinion, if you wanted to pull off something like that without confusing people you'd also have to establish a new prevalent mapping of the globe, with Europe/Africa far to the "west" and the Americas on the far "east" side.  Of course, established terms, like far-east, western society, etc. have to be recoined as well.


Deriving the Modigliani--Miller Theorem In the Wikipedia article on the Modigliani--Miller theorem, it states two propositions. (It gives the cases of with and without taxes. Here I'll just focus on the case without taxes.)
The first proposition is that the value of an unlevered firm is the same as a levered firm. Given the assumptions, this is clear from the discussion:

  To see why this should be true, suppose an investor is considering buying one of the two firms U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does. The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.

However, here I am asking about "Proposition II:"

  $$r_E(Levered) = r_E(Unlevered) + \frac DE (r_E(Unlevered) - r_D),$$
  where

  $r_E$ ''is the required rate of return on equity, or cost of equity,''
  $r_D$ ''is the required rate of return on borrowings, or cost of debt,''
  and $\frac{D}{E}$ ''is the debt-to-equity ratio.''

The article states that the "formula is derived from the theory of weighted average cost of capital (WACC)." (See a related question here.) My question is this: how can we arrive at this result from WACC?

The first equation can be written as:

$$ r_E(Levered) = \frac{E+D}{E}r_E(Unlevered) - \frac{D}{E}r_D $$

Then, isolating the unlevered return gives:

$$ r_E(Unlevered) = \frac{E}{E+D}r_E(Levered) + \frac{D}{E+D}r_D$$

And this is the WACC.


Difference between the minimum selling price and the actual selling price I'm actually wondering what the correct name for the difference between the minimum selling price and the actual selling price is?

Let's say I'm having an product A. After all calculations(costs, planned profit, possible discounts,...) I can sell my product for 7€(my minimum selling price). On the market I can offer it for 10€. In this case im making an extra margin of 3€.

That means: extra Margin(3€) = actual Selling price(10€) - minimum selling price(7€)

Now I'm wondering what the correct name of this extra margin(3€) is?

I already thought it could be the contribution margin, but isn't this something different?

Sale price minus cost is profit per unit or marginal producer surplus. It's not quite clear what this "planned profit" means. If you're including opportunity costs in this amount (I could spend my time working another job making X, so if I instead spend my time making Product A, I want to make at least X doing so), those are part of the costs. If you literally mean "How much I think I'll profit from this", that has no economic meaning.


Help with financial capital inflow I'm studying for an economics exam and am stuck on this question:

If consumers in Vietnam increase their demand for products that are manufactured in Mexico, which of the following will occur?

A. The supply of Vietnamese dong will decrease.

B. The Vietnamese dong will appreciate.

C. Mexico's financial capital inflow will increase.

D. The Mexican peso will depreciate.

E. Vietnam's financial capital inflow will increase.

The answer is E. Can somebody explain? From my understanding, since Vietnamese dong are going to Mexico to buy those manufactured products, there would be net capital outflow from Vietnam, not inflow. Thank you!

I don't like the question because money movement is a current flow and it is also capital flow.

Anyway, the current outflow (to pay for the goods from Mexico) will be matched by a capital inflow. Mexico will use the increase in earnings to buy goods and services and investments in Vietnam. If we assume proportions are constant before and after the increase in earnings, then some amount of the increase will go into investments in Vietnam. Investments are capital.


Why is common stock included in bank capital? Capital represents the loss-absorbing capacity of a bank. I do not understand how shares issued in the past provide any indication of a bank's loss-absorbing capacity. I understand that common equity does not carry with it an obligation to repay, but it is not like banks can "draw down" on the value of its outstanding share in the event it experiences severe losses.

Equity/stock/capital does absorb losses. Here is a stylized example:

Ann creates a new bank. She puts in \$100 of her own money. This \$100 is her equity/stock/capital.

She borrows another \$900 (debt). (For simplicity we'll assume that all debt/loans have an interest rate of 0%.)

She has a total of \$1,000. She lends all of it to Bob.

A little later, Bob is able to repay only \$925. So, \$75 was lost on this loan.

This \$75 loss is entirely absorbed by the equity/stock/capital, which now falls from \$100 to \$25.


Equity and increase in sales I am reading this book shoe dog by phil knight. There is mention of equity in one  of the pages where the banker says *Your rate of growth is too fast for your equity *  It is very confusing. From what i have read from google i understand that Equity= Assets - liabilities. Moreover , Assets increase when the sales of a product increase.I want to know how the sales increase is dangerous for equity. There is this excerpt from the book.]1

Here is a reason why growth of a firm's sales could be too fast.  Suppose the volume of sales of the firm's product is $N$ in period 0 and $N+\Delta$ in period 1. Suppose also that each unit of product sells for $P$, and the cost of the inputs for one unit is $cP$ where $0 < c < 1$.  Thus the profit per unit is $(1-c)P$, and total profit is $N(1-c)P$ in period 0 and $(N+\Delta)(1-c)P$ in period 1.  So far that looks satisfactory.

But suppose further that (as is quite possible) there is a lag of one period between purchase of the inputs to produce a unit and sale of that unit.  To keep things simple, assume there is no credit allowed on either purchases or sales, so that the cash outflow for the purchases occurs one period before the cash inflow from the associated sale.  Then in period 0 the cash inflow for period 0 sales is $NP$ while the cash outflow for the inputs for the period 1 sales is $(N+\Delta)cP$.  Thus net cash inflow in period 0 ($NCI_0$) is given by:

$$NCI_0 = NP - (N+\Delta)cP = [N - (N+\Delta)c]P$$

The condition for this to be positive is:

$$N > (N+\Delta)c$$

which is equivalent to:

$$ \Delta < \frac{N(1-c)}{c}\qquad(A)$$

If sales growth is sufficiently fast that relation $A$ does not hold, then, despite the firm being profitable, there will be a net cash outflow in period 0, and if such sales growth continues over many periods, there will be a continuing net cash outflow.  The question then is whether the firm can finance the cash outflow for long enough to see it through until a time when sales become more stable so that its profits begin to be realised in cashflow terms.

Finance could in principle be via either equity or debt.  However, a growing firm is unlikely in practice to be able to obtain debt finance unless the lender is satisfied that the firm's owners are risking their own equity to a sufficient degree.  It is possible therefore for a firm's sales to be growing at a faster rate than it is likely to be able to finance given its level of equity.


Solving for Pareto Efficient Utility Possibility Frontier using constrained optimisation The economy is a one-good two individual endowment economy in which individual $i’s$ preferences are given by $_(_)=_$, for ∈1,2, and the feasibility constraint on the amount of x available is $_1+_2=ω$.

We need to show that the equation of the utility possibilities frontier for this economy is $_2=ω−_1$

Setting up the Lagrange constrained optimisation function.

Max $U_1(x_1)= x_1$ subject to:
$$U_2(x_2) = x_2$$
$$x_1 + x_2 = w$$

$$L = x_1 + \lambda x_2 + \mu (w - x_1 - x_2)$$
$\lambda$ shows how small changes to $x_2$ affect $U_1$, and $\mu$ shows how small changes to the amount of x available in the economy affect $U_1$

Partial differentiation of all variables:

$$\frac {\partial L}{\partial x_1} = 1 - \mu = 0 \rightarrow \mu = 1$$

$$\frac {\partial L}{\partial x_2} =  \lambda - \mu = 0 \rightarrow \mu = \lambda$$

$$\frac {\partial L}{\partial \lambda} = x_2 = 0 $$
$$\frac {\partial L}{\partial \mu} = w - x_1 - x_2 = 0$$

$x_0 = 0$, so the Pareto Efficient point found involves the allocation where individual 1 receives all of x.

$\mu = \lambda$,so individual 1 is affected equally by changes to $x_2$ and by how much x is available in the economy.

I'm unsure how to conclude this, demonstrate that $_2=ω−_1$, or if I'm even on the right track. Could anybody please advise?

Also any feedback on my use of LaTex is also welcome, very new to using it.

The first constraint you have, $U_2(x_2)=x_2$, is not really a constraint.

Since points on the UPF is Pareto optimal, you should instead have $U_2(x_2)\ge \bar u$ or $x_2\ge \bar u$ as a constraint, where $\bar u$ is some arbitrary level of utility for individual 2 on the UPF. In other words, given that we're keeping individual 2's utility to be no lower than $\bar u$, how best can we make individual 1's utility as high as possible with the available resources.


Screening theory: Bundling problem Consider an example of a risk-neutral seller that has two distinct indivisible goods for sale. The seller wants to maximize the expected revenue. The buyer's utility is $$I_av_a+I_bv_b-t,$$ where $I_a,I_b$ are indicators of whether the buyer gets good $a$ or $b$ and $t$ is the monetary transfer. Both $v_a,v_b$ are the types of the buyer (willingness to buy the good).
Let $v_a$ and $v_b$ be i.i.d. draws from a uniform distribution on $[0,1]$.

Assume a seller quotes three prices: $p_a,p_b,p_{ab}$, where $p_{ab}\leq p_a+p_b$. What is the optimal choice of $p_a,p_b,p_{ab}$?

The textbook (T. Borgers, An Intro to the theory of Mechanism Design) I am studying simply says that this is a simple calculus problem and gives the solution as $$p_a=p_b=2/3\\
p_{ab}=\frac{4-\sqrt{2}}{3}.$$

I am not sure how to do that. I'd appreciate any help or hint. Thank you.

Consider the incentive and participation constraints. That is, look at which buyer types would choose which option.

A buyer does not buy anything if
$$v_a < p_a \quad v_b < p_b \quad v_a +v_b < p_{ab}.$$
A buyer chooses only good $i \in \{a,b\}$ with $j\neq i$ if
$$v_i - p_i \geq 0 \quad v_i - p_i \geq v_i + v_j - p_{ab}.$$
Otherwise she chooses the bundle.
With the assumption of independent uniformly distributed values, it then requires some caution to derive the probabilities of each event.

Finally, you want to maximize your profit given by
\begin{align}
(1-p_a)(p_{ab}-p_a) p_a + (1-p_b)(p_{ab}-p_b) p_b + \\
\left( \frac{(p_a + p_b - p_{ab})^2}{2} + (1 - p_a) (1 + p_a - p_{ab}) + (1 - p_b) (1 + p_b - p_{ab}) \\
- (1 - p_a) (1 - p_b) \right) p_{ab},
\end{align}
which is super annoying, but Mathematica suggests that your solution is correct.


Why do many papers not write the actual utility function? Why do many papers write the intertemporal maximisation problem for the household as
\begin{equation}E_{0} \sum_{t=0}^{\infty} \beta^{t} U\left(C_{t}, N_{t}\right)\end{equation}
and then do not write the actual utility function that could be something like
\begin{equation}U\left(C_{t}, N_{t}\right)=\frac{C_{t}^{1-\sigma}}{1-\sigma}-\frac{N_{t}^{1+\varphi}}{1+\varphi}? \end{equation}
Many papers state that they are following woodford (2003) for the microfundations. However also in that book there is no explicit utility function.
How is it possible to get the Euler equation or the IS curve without the actual functon?

This is done in order to make the solution more general. If you use specific utility function you prove your result only for that specific utility function and not for other ones.
You can still derive Euler equation. For example in a simple 2-period consumption model where agents maximize their expected utility
$$E[u]=E[\sum_{t=1}^T u(C)_t]$$
where $t=1,2$. Given this, the Euler equation would be simply:
$$u'(C_1)=u'(E[C_2])$$
you could insert some specific utility function into the problem. For example, you could use quadratic utility $E[u] = E[\sum_{t=1}^T C_t-\frac{a}{2} C^2_t]$ (as in Romer's Advanced Macroeconomic handbook) which would give you Euler equation $C_1=E_1[C_2]$ and continue using that but in the first case you prove your result for whole range of utility function whereas in the second case only for single 1 quadratic utility. In science we like to make results as general as possible so usually the first approach would be preferred.
In a typical IS-LM model you will actually want to substitute some utility function because for derivation of the model its important to know what optimal consumption $C^*$ is. However, you might not substitute it right away. You are completely vague about your citation of Woodford (2003) which could refer to multiple sources. For example, in his 2003 book Interest and Prices, on PP 145 where he builds New-Keynesian style macro model he actually makes explicit assumptions about consumption and in these macro models utility is usually kept simple such as $U=C-\frac{1}{\gamma}L^{\gamma}$ (See for example again above mentioned Romer's handbook).


Are taxes supposed to be calculated depending on the seller, or the buyer? We have a client that aims to ship his product internationally. (I think right now it mostly aims to be US and Canada, but the system would be open to people from all over the world to be bought)

Thing is, I can't seem to find a clear and definitive answer: Are taxes supposed to be calculated depending on the Buyers location, or the Sellers Location?

Examples:

If Seller is in the US, and Buyer is in BC, Canada; do I have to charge the BC GST/PST (about 12% on the price)?

If Seller is in QC, Canada and buyer is in the US, do I have to charge TPS/TVQ (about 15%)?

Also, in this case, there's a physical product being shipped, this isn't a webservice or something like that.

The most generic answer is that governments can basically tax whatever they want, according to whatever rules they choose to invent.  I have seen situations in which goods or services have been taxed by both the importing and exporting governments.

Having said that, there is a clear international norm that taxes such as VAT (the equivalent of sales tax in the vast majority of countries) apply in the importing country, after import duties (so you pay tax on the import duty and transport costs).

I don't know about the specifics of the US and Canada; I believe in particular US state sales taxes don't follow international norms.  NAFTA may also specify.


Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why? There is a market economy with technology given by:

$$Y = K^\alpha L^{1-\alpha} \tag{1}$$

Firms behave competitively and input prices are:

$$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})^{1-\alpha} \tag{2} $$

$$w = (1-\alpha) K^\alpha L^{-\alpha} = (1-\alpha) (\frac{L}{K})^{-\alpha} \tag{3}$$

I can solve for $\frac{L}{K}$, and obtain the following expression:

$$r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}} \tag{4}$$

Which is a function that links the prices independently of the inputs combination.

1) Is this correct?

2) What is the economic interpretation of the last result?

A clarification: The values $w,r$ are not independent of input combinations: you have already solved for $K/L$.

1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.

2) If there is a profit maximizing pair $(K,L)$, then for all $\alpha \in \mathbb{R}_+$ $(\alpha K,\alpha L)$ will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS).

A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence $r$ and $w$ will only determine the ratios of $K$ and $L$, not their level.


TFP estimation as residual and normality of errors Economists normally calculate TFP as the residual of a regression on output against K, L, and maybe other variables.

However, a well behaved regression is expected residuals to be normal (https://en.wikipedia.org/wiki/Normality_test). This is at odds with a situation where TFP increases over time, as in real economies.

Is then the method of residuals flawed? What are some adjustments made to overcome this problem? Maybe adding a trend?

TFP is not simply a residual value in a regression as you have assumed, it is called residual because it is observed rather than measured. That is what probably causes your confusion.

It is what is left after the Labour and Capital inputs have been accounted for. Normally you can break TFP down further into components of technological progress, scale effect and technical efficiency.

There is plenty of literature and sometimes conflicting views on the measurement of TFP and growth accounting on the web. Perhaps the below links can serve as a starting point.

Measuring TFP growth and TFP overview


Binary-continuous choice model in empirical consumer choices There are quite a lot empirical research based on discrete choice models, in which the consumer selects one of J alternative goods to maximize her indirect utility. The key assumption of these models is that there is an idiosyncratic utility component that follows the extreme value distribution.
The problem I am facing is different. I can observe a large number of individuals' consumption of a single good. The values of these choices are continuous quantities, with a long right tail, and roughly 80% of these values are zero. Even if I can discretize the quantity, I don't think I can assume an indepdent utility term for each discretized choice. So a different modelling approach is needed.
I conjecture the choice problem to consist of two steps. In the first step, the consumer decides whether to consume the good. If yes, the consumer then chooses how much to consume. In the empirical IO literature, is there any established model that is used for such "binary-continuous" choices?

It may not be a 100% match (and the details are a bit fuzzy in my memory), but I believe the model would be similar to Ericson and Pakes (1995).
In their model, there are $N$ firms who make sequential choices

Whether to enter the market or not (in your case, to consume or not), which is a discrete choice.
How much to invest if they have entered (in your case, how much to consume), which is continuous.

This is a seminal paper in dynamic discrete choice so I feel there should be some papers along this line.


Does Dogecoin or Bitcoin have a more appropriate rate of monetary inflation? The supply of new Bitcoins and Dogecoins is depicted in the graph below.

The Bitcoin inflation rate is currently higher than the Dogecoin rate, but it reduces sharply to a rate close to zero. It eventually reaches 0% in 2140, after which there will be no new Bitcoins.

There are currently 100 billion Dogecoins. The Dogecoin inflation rate is fixed at an additional 5.256 billion coins per year for every year henceforth.

Which inflation rate is more appropriate for a general-use currency intended for use for transactions of all sizes? I realise this isn't necessarily the stated aim of either currency.

Answerers should assume that the inflation rates are sacrosanct and can't be altered in years to come.

Your coins do not have 0 inflation in the long run

First, we need to set straight that you have a critical error in your supposition: The rate of creation of money is not the only determinant of the inflation rate, see here.

Even if we ignore these equilibrium effects, is any of these coins superior in its rate of money creation?

Suppose hereinafter that the actual inflation rate mirrors the money creation rate.

The optimal inflation rate is something that is not clearly understood, but most Economists agree that it is somehow related to the current state of the economy.

Hence, an inflation rate that is independent of the state of the economy cannot be that great. It basically removes the central banking authority to manipulate the inflation rate to the greater good of society - but removing this power is also an explicit goal of many advocates of these coins.

I have answered few reasons for and again higher inflation here. These make a case for a positive inflation rate. However importantly, they make a case for positive unexpected inflation rate. If high inflation is expected, all these benefits fall down. What remains is the shoe-leather-cost of inflation, implying that the lower inflation rate, bit coin, would be weakly better.

Remember: This is based on the false supposition that the money creation rate is the only determinant of the  inflation rate.

For the optimal rate of inflation, see also

http://www.frbsf.org/economic-research/publications/economic-letter/1997/september/what-is-the-optimal-rate-of-inflation/
http://www.nber.org/papers/w16054


Is this formula for net present value correct? I have been presented with this formula for the net present value:
$$NPV  = I_o + \sum \frac{F_t}{(1+r+p_t)^t},$$
where $F_t =$ net cash flow for period $t$; $R=$ required ret of return; $I_o=$ Initial cash investment, and $P_t=$ inflation rate during period $t$.
Although I've been told it is an industry standard, I don't get it.

Why does rate of return increase with time? I assume it is constant - if I want 8% return each year, I assume it comes from inflation adjusted value, so the term should be out of the brackets
"Initial cash investment" pharsing suggest the value will be positive, but I assume it has to be negative, otherwise it wouldn't make sense to sum initial investments with inflows.

Any comment will be appreciated, thank you!

Q2 : Indeed the correct formulation is for initial investment to appear with a minus sign, since it is an outflow. This reflects the position of somebody who contemplates making the investment. Note also that you shuold check whether there exists some residual value of assets at the end of the operational period examined - this should be added (and discounted) together with the net cash flow of the last future period examined.

Q1 : the logic of the net present value calculation is to notionally compare the returns of the specific investment with some abstract alternative that pays a constant $r$ per period, as a percentage of value left to produce returns. It is as though we have a bank account from which we draw and to which we deposit.

Now think in reverse : in order to "have" amount $F_3$ (three periods in the future), you only need to deposit now in this account $F_3/(1+r)^3$, and leave it there to accrue interest.

Now, if the cash flows are calculated in nominal terms, you would want to deflate them, so as to express them in today's value/purchasing power. The real value of $F_3$ is $F_3/[(1+\pi_1)(1+\pi_2)(1+\pi_3)]$, where $\pi$ is the rate of inflation. So over all we have

$$NPV (F_3) = \frac {F_3}{(1+r)^3 \cdot (1+\pi_1)(1+\pi_2)(1+\pi_3)}$$

Now, if you assume a constant inflation rate (or an average - after all, $r$ is also essentially an average), then you can write

$$NPV (F_3) = \frac {F_3}{(1+r)^3 \cdot (1+\pi)^3} = \frac {F_3}{[(1+r) \cdot (1+\pi)]^3} $$

$$=\frac {F_3}{(1+r +\pi +r\pi)^3}$$

Usually the term $r\pi$ is considered negligible (when we look at "usual" required rate of returns, and when inflation is low, think something like $0.1\cdot 0.02 = 0.002$). So it is common practice to be dropped (both in industry and in academia), arriving at

$$NPV (F_3) = \frac {F_3}{(1+r +\pi)^3}$$

What you were given keeps the inflation rate time variant. This comes from the approximation

$$i_t \approx r_t + \pi_t$$

i.e. that the nominal interest rate equals approximately the real interest rate/rate of return plus inflation. So the nominal rate of return may increase with time solely due to inflation.


How do I change the base year of real GDP using the GDP deflator and nominal GDP? I'll use the US as an example. I have three data series

nominal GDP $(Y)$
real GDP in 2005 USD $(\bar{Y})$
the GDP deflator $(d)$, with 2005 as the base year, so $d_{2005} = 100$

I want to change the base year to 2000. Are these calculations accurate? I use the notation $\$_{t}$ for USD in year $t$ prices to help myself keep the units straight.

My goal is $\bar{Y}_{t} \ \$_{2000}$.
\begin{align}
\frac{d_{2000}}{d_{t}} \cdot Y_{t} \ \$_t
&= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000} \ \$_{2005}} \cdot \frac{\bar{Y}_{t} \ \$_{2005}}{Y_{t} \ \$_{t}} \cdot Y_{t} \ \$_t \\
&= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000}} \cdot \bar{Y}_{t} \\
&= \bar{Y}_{t} \ \$_{2000} \cdot \frac{Y_{2000}}{\bar{Y}_{2000}}
\end{align}

I think those are all the correct unit cancellations, but now I'm stuck with the unitless quantity $\frac{Y_{2000}}{\bar{Y}_{2000}}$, so I don't know how to complete the conversion.

Am I doing this right?

It's just real GDP in year t, times (deflator in year 2000 / deflator in year t).


How are cryptocurrencies reported in GDP? Apparently the total market value of cryptocurrencies is currently close to $118 billion.

Is any of this reflected in GDP figures over the years? For example, if I start mining bitcoin today, I will consume electricity and produce bitcoins. How does this affect GDP?

Mining cryptocurrency can be considered an export of computing processing time, which is paid in a foreign currency (e.g. BitCoin).

The GDP is a measure of production of wealth of a nation. Exports and imports are considered in every method of calculating the GDP. You can read more on how each country calculates their GDP in this Economist article.

It is important to note that the actual GDP for each country is an estimate: it is very hard to calculate accurately the GDP, as you can imagine it is hard to calculate all of the consumption information about a given economy (cash transactions, informal markets, etc.).
However, economists try to include every single expenditure via estimations: the UK includes estimates of drug sales and prostitution as an illustration of the attempts to be accurate. I believe cryptocurrencies could fall in the same category.

Additional reading on the guesstimates of the GDP here.


How did 'crisis' spread through the financial system in '07-'08? How did 'crisis' spread through the financial system in '07-'08? Specifically, what is the process by which a subprime mortgage impacts the health of the credit markets as a whole? An explanation in the context of securitization would be appreciated.

I understand that subprime mortgages are sold to special purpose vehicles but the process is muddled from there.

I don't need a mathematical explanation, a clear conceptual explanation would be perfect.

Thank you!

Essentially, there was a bank run in the repo market.

The repo market is where big institutions (like, say, Fidelity) deposit large amounts of cash for short periods of time with institutions like, say, Bear Stearns.  Bear Stearns, in order to guarantee that it won't run away with the money, allows Fidelity to hold some valuable assets belonging to Bear Stearns in the meantime --- say some bonds.

Now Fidelity comes to believe that these bonds are worth only, say, 80% as much as it previously believed, and therefore asks for more bonds by way of collateral.  So, at around the same time, does everyone else.  But Bear Stearns only owns so many bonds, so it can't satisfy everyone.  If you've ever seen "It's A Wonderful Life", you know what happens next.

That, at least, is Gary Gorton's story, and it seems very plausible and is widely endorsed.  Of course there are many competing stories as well.


Where can one find Beta values for stocks? Is there someplace specifically on the internet where one can find Beta values for stocks? I've been looking but apparently I'm looking in the wrong places. Tried WSJ stocks and Morningstar. Morningstar need to subscribe.

Google Finance and  Yahoo Finance are both sites which provide free information on Stock prices and their variables.

I prefer google as it is easier to use, but both sites are pretty good.


Insurance and Hirshleifer effect I am trying to understand the statmement that 'public information kills insurance opportunity' -- referred to as Hirshleifer effect. Does it (in general) lead to some undesirable outcomes? Could you explain that to me with a simple example? For me it is easy to form examples of trade breaking when there is asymmetric information (adverse selection for example) and that having an undesirable effects. But I am not able to fully grasp the meaning and consequences of Hirshleifer effect.

Thanks

UPDATE: Here is what I thought could be a simple example. Suppose there are two periods ($t_0, t_1$) and two possible states of nature ($s_1,s_2 $) that arise in period $t_1$. Suppose there are two agents. The first agent has an income stream that pays off only in state $s_1$ while, second agent's income stream pays off only in state $s_2$. Consumption happens only in $t_1$ and both agents prefer to consume in both the states than in just one (Inada condition as consuption goes to 0 is not satisfied) and more is always better. If at $t_0$, neither agent knows the state at $t_1$ then they might enter into a trade that ensures them positive consumption across both the states.

On the other hand, if, in $t_0$, state is already revealed publicly (in this example even privately to one agent) an agent who has a positive pay-off in $t_1$ would not be interested in any trade with the unlucky agent.

Is this way of thinking in the example correct?

Thanks for any comment


The existence of free lunch Is there a free lunch or something beneficial but with zero price? Is air in an unpolluted area free lunch? If so, the sentence "there is always a tradeoff between benefit and cost" is not in general true. Is this correct?

The concept of a free lunch is a philosophical/political phrase (not an axiom or law in economics). It was popularized by Nobel Prize winning economist Milton Friendman

The concept depends too largely on how you define free to make a general statement about the concept. For example, while Oxygen is readily available for your consumption from the atmosphere at a cost of zero, Friedman argued there was always an opportunity cost involved (you could be breathing ultra-pure oxygen from an oxygen tank to avoid pollutants in the atmosphere that will give you cancer) so technically there is a cost.


Macroeconomics and equilibrium Preliminary note: I have nearly zero economical background, I am a physicist and my question is probably very naive, but I would like a clear and simple explanation.

If I am not wrong, the global economy can be seen as an isolated system (ie we do not have any non-Earth economic exchanges). At any given time $t$, there is a given amount of money $M$ in the world. As worldwide economy is an isolated system, $M$ can vary only if it is created or destroyed within the system. To my knowledge, the only way to create money is by creating a debt at $t_{0}$ and when money is reimbursed at $t_{1}$, the money that was created out of thin air is destroyed. I see no problem when the interest rate is $0\%$, because globally the system stays balanced.

Now, if the interest rate is stricly positive, that means that the amount of money $|m_{1}|$ that will need to be reimbursed at $t_{1}$ is higher than the debt $|m_{0}|$ that was created at $t_{0}$. But the thing is that $dm = |m_{1}|-|m_{0}|$ does not exist in the total amount of money available worldwide at $t_{0}$, $M_{0}$. So for the debt to be reimbursed with the interests, $dm$ will need to be created. And the only way to create money... is by creating a debt... with interest. So the system diverges very rapidly with an ever growing debt.

The other way I can see to make the system stable is by inflation/deflation: if $dm$ exactly corresponds to the inflation/deflation rate, then the system can remain at equilibrium without an ever growing debt. The last way I can see to remain at equilibrium, is by crisis so that $dm$ is never reimbursed.

I would like an explanation in simple words about what is wrong about this naive view.

The world economy is a closed system as regards economic exchanges, but it is not a closed system as regards productive resources: earth as a natural system still has resources outside the human economic system that are continuously transformed by willful human activity and enter the economic system. Also, human time (enhanced with productive know-how) is something that continuously flows into the system as a productive resource.

The issue with the quantity of money therefore, is not its absolute level and magnitude, but its quantity relatively to produced output. A small village with a monetized economy, needs a certain amount of money to function economically. A large country needs a much higher quantity of money for the same purpose.

When a person borrows an amount of money, he essentially borrows an amount of productive resources, raw (investment loan) or in finished/ready-to-consume form (consumption loan). And indeed, he undertakes the obligation to return a higher amount that includes the interest.

If, when the time comes to repay loan+interest, he liquidates some already existing asset of his, then we do not really have additional money creation to cover the interest. If it manages to pay loan+interest by the output he produced in the meantime through his activity, then the additional money that needs to be created in order for the monetary transaction to be completed, "makes no harm" since it matches and reflects underlying real value that has already been created.

Certainly, "value" is not objective, and we have the price system that fluctuates, creating defaults, bankruptcies, hyperinflations, sudden re-distributions of wealth, etc. But in the very large picture you are looking at, these are fluctuations and deviations around the structure delineated above.


Does savings always equal investment? Are there any cases where savings could be removed from the economic system altogether and break the balance? Assuming a closed economy, does savings always have to equal investment?
If an individual produces a certain amount of a resource and saves it, that becomes part of the savings of the economy. However, if the resource is subsequently destroyed, it is then removed from the economy. In that case, the savings have increased but the investment hasn’t?
Does destroying a resource count towards investment? Is that an edge case we don’t consider at all? Is this any different from the way investment takes resources away from an individual and puts it into an economy?

Does savings always equal investment?

If we are talking about national accounts then yes. Output of closed economy is given by:
$$Y = C+I+G \tag{1} $$
Private saving $S_p$ is by definition is disposable that is not consumed (see further explanation in Blanchard et al. Macroeconomics an European Perspective pp 55) so:
$$S_p= Y- T-C \tag{2}$$
Public saving $S_g$ is by definition equal to difference between taxes $T$ and  gov spending $G$ (see ibid) so we have:
$$S_g= T-G \tag{3}$$
Substituting 3 into 2 (by solving for T) and then into 1 (by solving for Y) gives us:
$$S_p+S_g + G + C= C+I+G \tag{4} $$
Now just by simple algebra we can see $G$ and $C$ are eliminated so we are left with:
$$S_p+S_g =I$$
So total saving (public and private), must be at all times equal to investment. The above is pure accounting identity, it simply must hold the same way as it is not possible that in some company assets are not equal to liabilities and equity.

However, if the resource is subsequently destroyed, it is then removed from the economy. In that case, the savings have increased but the investment hasn’t?

No. You need to realize that $C, I, G, Y  $ or $S$ must be defined over some time period like a day, month, year, decade and so on. Let us assume that we are talking about yearly period for a simplicity sake. For example suppose that in 2010 goods worth of 1000e were produced, people wanted to save 200e but in the same 2010 100e  were destroyed (gov is assumed away). Well the GDP gets calculated at the end of the time period so if we would look inside the time period before destruction GDP would look like $Y=1000$, $Y-C=800$ $S=I=200$ but after destruction it would look like  $Y=900$, $Y-C=800$ and $S=I=100$. You see that it would all perfectly balance and overall what is reported at the end of the year is  $Y=900$, $Y-C=800$ and $S=I=100$.

Does destroying a resource count towards investment?

No, but it will no longer count toward output as well. Investment will be lower but output for the same period will be lower by equal amount making it all equal.
To give you an intuition, you should realize that $Y$ output/income must always be equal all spending $C+I+G$ (for closed economy), if you lower spending on  $I$ (because part of it is destroyed and thus no longer counts), $Y$ has to drop as well.  What you are saving in real terms is not money but output -that is what matters, you are consuming output, investing output (which is equivalent to saving output as proven above) or government is consuming output - it is all just breakdown of what happens with output Y, saving part of it does not make it no longer output.

Is this any different from the way investment takes resources away from an individual and puts it into an economy?

Yes of course. Investment is basis for capital formation which in turn expands the societies production possibility frontier (see discussion of this in ibid ch 14).
Destruction of those resources will lower the total investment, hence ceteris paribus negatively affect future output (however, note here everything is expressed in real terms not nominal terms i.e. destroying money would not have the same effect as that would just change nominal value). However, as explained above it also lowers present income so national accounts perfectly balance each other.


What have been some trade-offs that the U.S. has made? In Thomas Sowell's book titled, "Basic Economics", he says the following:

"Unmet needs" are inherent in these circumstances, whether we have a capitalist, socialist, feudal, or other kind of economy. These various kinds of economies are just different institutional ways of making trade-offs that are inescapable in any economy.

He has been talking about what scarcity means and what it implies -

It means (Scarce) that what everybody wants adds up to more than there is. What this implies is that there are no easy "win-win" solutions but only serious and sometimes painful trade-offs.

In a real world context, what would be some big trade-offs that  the U.S., specifically, has made? And, what have been some consequences or maybe some pros of those decisions?

The reason I am asking this is because I can't really 'picture' what exactly a trade-off is.

Thanks for your time!

It might be easier to grasp the idea of a tradeoff by framing the concept at a less-than-national scale.

At a personal level, you might think of a tradeoff as giving up a little bit of one thing so you can have a little bit more of another. So, maybe you save your lunch money for a few weeks so you can get new shoes. The thing that is scarce here is money. You save your money (you accumulate your scarce resource) by forgoing lunch (the sacrifice) and then buy new shoes (the gain). The tradeoff is the lunch/shoes. You were a bit hungry so that could dress better and impress the ladies.

This is a bit naive but it illustrates the idea here: a tradeoff is just achieving a balance between desirable but incompatible/mutually exclusive things.

A better example that actually answers your question (all nations face this) is the unemployment/inflation tradeoff mapped by the Phillips curve. Here, the tradeoff is the rate of unemployment a nation must endure relative to the amount of inflation a nation must endure.

As you can see in the image, reducing inflation comes at the cost of increasing unemployment and reducing unemployment bears the cost of increased inflation. Thus, a nation must optimize its outcome by striking a balance between the two.

That should be enough for you to think of what tradeoffs America faces. A good place to look might be at national budgeting decisions. The recent financial crisis certainly forced some big decisions. If you understand the concept it is relatively easy to identify tradeoffs.


Sum of residuals and Matrix calculus (step by step) Write the sum of residuals in matrix form.

Attempt: $(y-X\boldsymbol{\beta})^T(y-X\boldsymbol{\beta}) = y^Ty-\boldsymbol{\beta}^TX^Ty-y^TX\boldsymbol{\beta}+\boldsymbol{\beta}^TX^TX\boldsymbol{\beta} = y^Ty-2y^TX\boldsymbol{\beta}+\boldsymbol{\beta}^TX^TX\boldsymbol{\beta}.$

  Minimize this and verify that $\hat{\boldsymbol{\beta}} = (X^TX)^{-1}X^Ty$.

Attempt: $\frac{\partial SSR(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -2y^TX+2X^TX\boldsymbol{\beta} = 0 \rightarrow X^TX\boldsymbol{\beta}=y^TX \rightarrow \hat{\boldsymbol{\beta}} = (X^TX)^{-1}y^TX = (X^TX)^{-1}X^Ty.$

I haven't studied matrix calculus, so I am a little bit confused about the step in calculation. In the first problem, could you show me which rule is used for the first equality in the equation. And in the second problem, I am a little confused about the first step of differentiation.

I really appreciate if you show the calculation step by step.

Let $\mathbf A,\mathbf B$ be $m\times n$ matrices (or vectors is either $m$ or $n$ is $1$),
\begin{align}
(\mathbf A+\mathbf B)^T&=\mathbf A^T+\mathbf B^T \tag{1}\\
(\mathbf A\mathbf B)^T&=\mathbf B^T\mathbf A^T \tag{2}
\end{align}

Thus,
\begin{align}
(\mathbf y-\mathbf X\beta)^T(\mathbf y-\mathbf X\beta)&=(\mathbf y^T-(\mathbf X\beta)^T)(\mathbf y-\mathbf X\beta)\\
&=(\mathbf y^T-\beta^T\mathbf X^T)(\mathbf y-\mathbf X\beta)\\
&=\mathbf y^T(\mathbf y-\mathbf X\beta)-\beta^T\mathbf X^T(\mathbf y-\mathbf X\beta)\\
&=\mathbf y^T\mathbf y-\mathbf y^T\mathbf X\beta-\beta^T\mathbf X^T\mathbf y+\beta^T\mathbf X^T\mathbf X\beta \\
&=\mathbf y^T\mathbf y-2\mathbf y^T\mathbf X\beta+\beta^T\mathbf X^T\mathbf X\beta \tag{3}
\end{align}
where line 1 uses $(1)$, line 2 uses $(2)$, and the last line follows from the fact that $\mathbf y^T\mathbf X\beta$ and $\beta^T\mathbf X^T\mathbf y$ are both $1\times1$ scalars.

Let $\mathbf a,\mathbf x$ be $n\times 1$ vectors and $\mathbf A$ a symmetric $n\times n$ matrix. Matrix calculus has the following rules:
\begin{align}
\frac{\mathrm d\ \mathbf a^T\mathbf z}{\mathrm d\ \mathbf z}&=\mathbf a \tag{4} \\
\frac{\mathrm d\ \mathbf z^T\mathbf A\mathbf z}{\mathrm d\ \mathbf z}&=2\mathbf A\mathbf z \tag{5}
\end{align}

Applying these to differentiating $(3)$ with respect to $\mathbf\beta$, we get the first order condition:
\begin{equation}
-2\mathbf X^T\mathbf y+2\mathbf X^T\mathbf X\beta=0
\end{equation}
Solving for $\beta$, we get the usual OLS formula.


Single Peaked Preferences and Condorcet Cycles How can you prove the theorem that Single peaked preferences imply that there are no Condorcet cycles

Single-peakedness of preferences is not enough to guarantee the existence of a Condorcet winner, a position that cannot be defeated in majority voting (hence preventing cycling). You would also need that the set of alternatives to admit a linear order (and single-peakedness is with respect to this order).

With a linear order on alternatives and a profile of single-peaked preferences, it is easy to argue that the "median voter", a position that divides the set of voters into two subsets of equal size based on whether their bliss points are on the left or right of the median position, is always a Condorcet winner. For a more rigorous presentation, I'd refer you to Proposition 21.D.1 and its proof in the MWG textbook.

To see why the linear order on alternatives is necessary, consider the following figure (taken from Wikipedia):

Here, the space of alternatives is 2-dimensional, and there is no natural way to define a linear order on $\mathbb R^2$. Voters' preferences are still single-peaked in the sense that each voter has only one bliss point. As a result, we cannot rule out cycling (non-existence of Condorcet winner) as is shown in the example.


Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ] A consumer with a utility function given by $u(x_1, x_2) = \sqrt{x_1} + x_1x_2$ has an income of $100$. The unit prices of $x_1$ and $x_2$ are $4$ and $5$, respectively.
(a) Compute the utility-maximizing commodity bundle, if consumption must be nonnegative.
The necessary condition for this optimization is that
$$\partial L/\partial x_1 = \frac1{2\sqrt{x_1}} + x_2 -4\lambda_1+\lambda_2 = 0$$
$$x_1 - 5\lambda_1 + \lambda_3 = 0$$
$$100 - 4x_1 - 5x_2 = 0$$
$$x_1 \ge 0, \quad x_2 \ge 0$$
There are two cases we need to consider $\{x_1 >0, x_2 >0\}$ and $\{x_1 >0, x_2 =0\}$. For the first case, if I solve the above system of equations, I have
$$100\sqrt{x_1} = 8x_1^{3/2} - 5.$$
In this case, how can we obtain the bundle $(x_1^*, x_2^*)$?

The equation $100\sqrt{x_1} = 8x_1^{3/2} - 5$ is cubic in $\sqrt{x_1}$.  Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $5$, enabling division by $\sqrt{x_1}$ yielding:
$$100 = 8x_1$$
The first approximation is then $x_1 = 100/8 = 12.5$. Since $100\sqrt{12.5} \approx 354$ is much greater than $5$, it can be inferred that the exact solution will be very close to $12.5$.  So try successive approximation, perhaps starting with $x_1=12.6$.


Correlation between CPI and GDP What is the relationship between GDP and CPI(consumer price index)? My thinking is that if the CPI increases, this means that the market basket cost has increased, therefore, the consumer spending has increased. If the consumer spending increases, then GDP increases. However, depending on the consumer preferences, CPI can either underestimate or overestimate the cost of living. So from here, I cannot draw consequences about how CPI affects GDP or vice versa. Can someone please explain this? Is my thinking correct? Can there be any additions to these?

Any help would be appreciated!

Thanks in advance!

http://econsmalaysia.blogspot.my/2013/05/gdp-deflator-inflation-consumer.html. Here read this. Its almost a non-related type of measurement
From the article:

The differences in calculation are quite substantial and meaningful – they’re not looking at the same thing. In fact, calculating the GDP deflator is fairly convoluted as these things go – first tabulate all goods and services in current prices, then measure the same goods and services in prices prevailing in the base year, then calculate the ratio of current price production to constant price production. This ratio (multiplied by 100) is the GDP Deflator index, from which growth rates (inflation) can be calculated.
The CPI on the other hand, is based on changes in prices only, as the volume of goods consumed by a “representative” (average) household is taken to be fixed. The index is just a weighted average of the changes in prices across the same basket of goods.

As an example, below are the GDP deflator and CPI for Malaysia, which show weak correlation

The key thing to understand is that while both can be affected by broad macro policy (like loose money), they do not neccessarily always drive each other, as the GDP deflator could be driven by a surge in export commodity pricing that is largely captured by foreign firms with little corresponding increase in pay to local workers and consequently little increase in CPI.
The relationship would have to be examined in terms of a single economy because the weight of outside factors confounds the relationship between the CPI and GDP


What are some examples of modern economic research that integrate economics and computer science? I am entering my first year of graduate studies in Economics this fall and I'm curious about which areas of economic research utilize various fields of computer science (e.g. Machine Learning, Graph Theory, etc.) to study economics. More specifically, how are recent advancements in the field of computer science being used to answer economic questions that were previously difficult or impossible to answer?
Thank you in advance for any responses.

I can try to give a bunch of examples about specific branches in economics that have been reshaped by computer science over the last decade, but I will leave this to the ones that are more knowledgeable, and far better writers.

That being said, I really enjoyed Hal Varian's take away of it:

https://www.aeaweb.org/articles?id=10.1257/jep.28.2.3

And this video where Susan Athey elaborates upon the role of machine learning in the study of causal effects and their connection with impact evaluation literature:

https://www.youtube.com/watch?v=VeTl4mIdqWA

Good luck in your studies!


Why does the gold standard require countries to keep exchange rates fixed? The gold standard required countries to use monetary policy to keep
exchange rates fixed and thus to allow prices, output, and employment
to vary as required by the movements of gold and the country’s
exchange rate.  (A history of the Federal Reserve, volume 1, 1913–1951.  Allan H. Meltzer, 2003, The University of Chicago Press. ISBN 0-226-51999-6.  Introduction, page 5.)

What established fact is being invoked here?  Can you cite a reference
that provides instruction about this result?  Thank you.

This is required just due to the simple definition of what gold standard is. For a monetary arrangement to qualify as gold standard money has to be convertible into gold. Hence central bank or some other government body must be always prepared to convert notes into gold. That is part of both the dictionary and economic definition of how it works.
To see why this requires fixed exchange rate lets say that this gold conversion is set in US such that $\\\$1$ converts into $1$ gram of gold. Now suppose EU would set its conversion rate to $1e = 2g$ of gold. If the exchange rate would be anything else than $E=\frac{\\\$}{e}=2/1$ the gold standard would eventually have to collapse in long run.
For example if exchange rate $E$ would be $1$ (i.e. $1EUR=1USD$) then everyone would take their dollars convert them to euros demand gold from European Central Bank to the point when either it would left without any gold to support the gold standard or it would change the exchange rate. If the exchange rate would be higher than $E=2$ opposite would happen. Consequently, under gold standard maintaining fixed exchange rate is necessary otherwise it cannot survive due to the fact that under gold standard notes must be convertible into gold.


Trouble at differentiating a consumption function everyone.
This is kind of a basic question, but I have a weak background on calculus and I already tried to figure this out by myself for a few hours without success. So, I am finally appealing on you - I even feel embarrassed about asking such a basic question among all of these complex questions in the forum.
Let's get to it: in the middle of a chapter about aggregate demand and IS curve in the textbook "Macroeconomia", by Mario Henrique Simonsen, there is the statement of this consumption function:
$(1)$ $C = aA + bY_d$   with $a>0$ and $b>0$
(C represents consumption, A represents non-human wealth and Yd represents disposable income)
Then, as keynesian theory takes A as given in the short-run (so it is treated as a constant), there is this differentiation of A in respect of t:
$(2)$ $\frac{dA}{dt} = Y_d - C = (1-b)Y_d - aA$
And there is my problem. Mathematically, I don't know how $\frac{dA}{dt} = Y_d - C$.
My train of thought is to differentiate the consumption function like this:
$\frac{dC}{dt} = a \frac{dA}{dt} + b \frac{dY_d}{dt}$
Then, as $\frac{dA}{dt} = 0$:
$0 = b \frac{dY_d}{dt} - \frac{dC}{dt}$
But it is pretty different than equation (2). What are my mistakes? Thank you.

This does not seem to have anything to do with calculus.
The idea is that the income not consumed $Y_d - C$ is saved (usually denoted by $S$).
This saving is then lent out to companies (via banks) who invest it (usually denoted by $I$), and the accumulated capital is used in production. I am guessing this is denoted by $A$?
And the change of $A$ in time is the investment (assuming there is no amortization).
Hence
$$
Y_d - C = S = I = \frac{dA}{dt}
$$


Mix of two ads types Question still unanswered

Ads in the boring business magazine are read by 300 lawyers and 1,000 M.B.A.s. Ads in the consumer publication are read by 250 lawyers and 300 M.B.A.s. If Harry had \$4,200 to spend on advertising, if the price of ads in the boring business magazine were \$700, and if the price of ads in the consumer magazine were \$350, then the combinations of recent M.B.A.s and lawyers with hot tubs whom he could reach with his advertising budget would be represented by the integer values along a line segment that runs between the two points

a.
(3, 600, 0) and (0, 7, 200).

b.
(3, 000, 3, 600) and (1, 800, 6, 000).

c.
(0, 3, 600) and (1, 800, 0).

d.
(3, 600, 4, 200) and (1, 800, 7, 200).

e.
(2, 400, 0) and (0, 6, 000).

My attempt was to find a budget line which is $700b+350c=4200$, where b stand for business magazines and c for consumer ones. But i do not know how this would translate into the answers that are given.

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

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

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

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

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

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

Cheers.


How are ties broken in a second price sealed bid auction? Say there are two people who evaluate an object at 5 dollars and a third who evaluates it at 4 dollars playing a second price sealed bid auction. The first two bid 5 dollars and the third bids 4 dollars. How is the tie broken and how much do they pay?

The winner pays the second highest bid, which is 5 dollars, however who the winner is depends on the tie-breaking rule. There are multiple rules one could use. In most cases it is assumed that the winner is selected randomly from among the people who placed the highest bids. But you could also select the winner from this group alphabetically or by checking who submitted their bid first.

In case of a Vickrey auction in equilibrium when multiple players place the same highest bid all such players are indifferent between winning and losing, so the exact tie-breaking rule is usually not very important, it does not affect the players' equilibrium payoff nor the seller's revenue.


Accelerator and Multiplier effects at full employment I have a question regarding the multiplier and accelerator effects. The way I see it, the value of the multiplier tends to be lower as the economy approaches full capacity. I would therefore expect growth to be slower as full capacity is reached. The accelerator effect works best when there is no excess capacity as an increase in aggregate demand would mean producers use their already existing idle fixed capital.

However the accelerator is based on past increases in consumption and output; for the investment to increase, growth must therefore increase at an an increasing rate. If at long run full capacity, growth is slower due to a lower multiplier value and the inability of firms to expand output, how can the accelerator effect come into existence???

I am sorry that this was long but I am unsure as to whether I have made a mistake early in my "chain of reasoning"...

Grigory, I am glad you are skeptical of this 'theory'.

I am not sure I understand you line of argument correctly, but given the assumptions it seems to be correct. Accelerator cannot be sustained in long-run due to fixed factors of production (think land, labor force, etc). Accelerator theory is the sense you are presenting it is not present in modern macroeconomic literature or graduate curriculum.

Let's then turn our attention to the assumption that 'multiplier' at full capacity. The old Keynesian theory indeed assumed that the multiplier is higher when there is high unemployment. Indeed, this was a reason for advocating for public spending during recessions.

However, modern evidence do not seem to support this view. For example Ramey & Zubairy (2018, Journal of Political Economy), show that the multiplier does not vary significantly with the business cycle if one controls for expectations. Keep in mind, however, that those things are notoriously hard to measure. An ideal experiment, that would allow us, economists to measure the multiplier, would consist of an unexpected debt-financed government spending in the absence of central bank. You can see that all of these assumptions are typically violated (taxes and central bank's action matter in particular).

So the bottom line is, always take simple, undergraduate-level models with a pinch of salt. Some are logically inconsistent and most (if not all) are consistent with data.


What are some exceptional movies / documentaries on macroeconomics? I would like to know a few movies / documentaries that are based on macroeconomics

Princes of the Yen is a documentary that springs to mind.


Why are nations engaging in austerity when bond yields are historically low? In the UK and EU austerity programmes are rife. However, bond yields are also historically low, and have even been negative! At a time when debt is cheaper than ever, and below the accepted long term rate of inflation, why do governments not take this "free" money to maintain their states? Or, is it the austerity itself that is driving the rates?

UK bond yields:
http://www.tradingeconomics.com/united-kingdom/government-bond-yield
EU bond yields:
http://www.tradingeconomics.com/euro-area/government-bond-yield

Debt being cheaper than ever doesn't mean that it is optimal to borrow. Unless you plan to invest the borrowed money into projects that yield higher returns with certainty (we would call this arbitrage), any debt implies a tradeoff between the welfare of current versus future generations.

Perhaps the borrowing before to higher rates was non-optimal under the current government's perspective. I.e. now we place a higher value onto future generations versus current generations
Perhaps now in the slump, the government doesn't see investment opportunities that yield a sufficiently high return.
Perhaps the current government has a different prior on how high returns to debt are. Different governments (or the same government over time) can have a different prior (guess) on how important/productive government spending onto infrastructure, education, health, military etc. is. This also includes a prior on how large the Keynesian multiplier is, and how effective demand stimuli are. Then, even presented with the same investment opportunities at a
lower interest rate, a current government might chose differently
than the previous government.

This was all without political economy consideration. There are arguments on optimal spending that take into account how your current budget will affect the budget of future (different) governments. Have a look at Persson and Svensson (1989) to get a feel for this kind of political economy reasoning.

Note that the argument in the cited paper would go the wrong way (i.e., too much spending of a conservative government), but it shows you how optimal policies under a permanent ruler can differ from optimal policies in a democracy.


Why intuitively does Break-even point $= \dfrac{\text{Total fixed costs}}{\text{Selling price per unit} − \text{Variable cost per unit}}$? p. 100. Accounting: A Very Short Introduction  (2014).
I understand the green sentence beneath, but not the fractional formula for BEP containing CPU in red? What's the intuition for the latter?

Contribution
The ‘contribution’ is the amount that a particular product line helps
towards covering fixed costs and making a profit. The contribution is
measured as: Sales − Variable costs.  This can also be calculated per
unit of production. The contribution per unit (CPU) is:
$\color{red}{CPU = \text{Selling price per unit} − \text{Variable cost per unit}}$
That is, taking the selling price for one unit of production, and then
deducting all the variable costs for that one unit of production,
gives the contribution per unit.
Break-even point (BEP)
$\color{forestgreen}{\text{The break-even point is the volume of production at which the firm makes zero profit.}}$ Therefore, the break-even point is the level at
which all costs (including fixed costs) can be paid for. The formula
is:
$BEP = \dfrac{\text{Total fixed costs}}{\color{red}{\text{Contribution per unit (CPU)}}}$
So, above the break-even point, any extra items sold will generate
profit. Every unit that is produced and sold in excess of the
break-even point will generate profit of the amount of the CPU. The
total profit made is:
Units in excess of break-even point × CPU
If the firm does not reach break-even point, this implies that it has
not yet covered fixed costs and will therefore make a loss. The size
of the loss will be the number of units under the break-even point,
multiplied by the CPU.

The break-even point can be
derived using the formula given above, or by drawing up a chart such
as Figure 15.

Intuitively, the breakeven point is where the volume of sales results in the gross margin (based on the difference between selling price and variable costs per item) exactly covering fixed costs


Is sending fake resumes deception? Economists generally frown on deception. For example, the journal Experimental Economics does not publish papers which use deception. However, I know that famous economics studies have been published where people send fake resumes to employers (to detect racial discrimination). For example, Bertrand and Mullainathan (2004).

Does this mean that sending fake resumes is not deception?

Fake resumes are a form of deception in a manner of speaking, but the purpose of the deception is to reveal obfuscated bias in hiring practices. Not all journals take such a hard-line stance on this type of deception and as such papers can still be published.

For a more robust discussion on the main dimensions that determine the judgment of deception, I recommend Krawczyk (2013).


Trouble Understanding the Integral Underlying Random Utility Models Given a utility function, $U_{nj} = V_{nj} + \varepsilon_{nj}$, it makes sense that we can find the probability the decision maker $n$ chooses alternative $i$ as:
$$Pr(U_{ni} > U_{nj} \forall j \neq i)$$
$$=Pr(V_{ni} + \varepsilon_{ni} > V_{nj} + \varepsilon_{nj} \forall j \neq i)$$
$$=Pr(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)$$
I am having an issue understanding the next step though, where:
$$Pr(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)$$
$$= \int_{\varepsilon} \text{I}(\varepsilon_{nj} - \varepsilon_{ni} < V_{ni} - V_{nj} \forall j \neq i)f(\varepsilon_n)d\varepsilon_n$$
Specifically, I am having trouble wrapping my head around why the joint density function, $f(\varepsilon_n)$, comes into play. I understand that this is one of the crucial aspects to understanding discrete choice models, since the choice of this joint density function underlies what model is to be used. Hence, I would like to gain a better understanding of why it is in here. If anyone would maybe have an example or a dumbed-down explanation of why this integral is what it is, I would appreciate it.

This has nothing to do with any specific model. For any event $A$, let $I_A$ be the indicator function such that $I_A(\omega)=1$ if $\omega\in A$ and $I_A(\omega)=0$ if $\omega\notin A.$
Then $\mathbb{P}(A)=\mathbb{E}[I_A],$
and here the expectation is given in terms of a density function.


What factors affect the temporal discount rate individuals apply in decision making? Discount rate is rate at which actors will value something that occurs in the future at less than its value now.

A simple hypothetical test that would determine this would be:

'Would you rather have \$100 now, or \$100 in one year?'
'Would you rather have \$100 now, or \$150 in one year?'
'Would you rather have \$100 now, or \$200 in one year?'

And so forth, to determine just how much someone values money now, over money in the future.

Discount rates can explain why criminals for example, commit seemingly irrational crimes - the cost of potential imprisonment being discounted because it's an abstract outcome that occurs in the future, compared to an immediate gain now.

My question is - is there research into what factors affect and individual's discount rate?

Eg. things like age, education, income.

And in what circumstances was the discount rate being measured?

I think one can answer this question by looking at what affects saving rates.

Age

According to the life-cycle model of household savings*, young and old people save less than middle-aged people. Young people who are starting a new family have many expenses and a lower income due to lack of work experience. Thus, young people save less and value money more in the present (a high reservation intertemporal discount rate). Middle-aged people have a higher income and save towards their retirement (a low reservation intertemporal discount rate). Old-aged people retire and live off their savings, so they have a high reservation intertemporal discount rate.

Dependants

If you support more dependants, you will be shorter of money and will have a high reservation intertemporal discount rate.

Investment Opportunities

Better quality investments and financial institutions will improve the interest rate at which money can be saved, which increases the intertemporal discount rate at the intersection of the budget constraint line and the indifference curve of the individual.

Income

One would think that saving increases with income, but high-income countries actually save less than middle-income countries, but moreso than low-income countries (World Bank, data on Gross Domestic Savings (% of GDP)). So there is no obvious relationship with the level of income, but if the anticipated future level of income goes up, then people will save less (and consume more in the present) giving a higher intertemporal discount rate.

*Alberto Aldo and Franco Modigliani, "The Life-Cycle Hypothesis of Saving: Aggregate Implications and Tests," American Economic Review 53, no. 1, (March 1963), 55-84.


Linear Probability Model Instead of Logit in Fixed Effects Regression In our panel data analysis we estimated a fixed effects linear probability model (LPM) instead of a fixed effects logit regression because our sample size was quite small (600 individuals) and the fixed effects logit decreased our number of observations hugely (to less than 200 at times), while our LPM kept much more observations. Likewise, our logit wouldn't converge as we had to include region indicators and very few people moved such that we had issues of small numbers in cells in logit. Is there a stronger way to explain/justify this, or a preference for linear probability models over logits more generally in panel data with fixed effects? We completed our analysis in Stata.

FE logit requires the idiosyncratic errors to be IID across $i$ and $t$, quite a strong assumption. Also the regressors should be strictly exogenous, but it's the same for linear FE models. In your application, the fact that FE logit wouldn't converge will make a good argument against FE logit, and will satisfy some referees but not all.

An important drawback (or perhaps more like a "feature") of FE logit is that partial effects (e.g., average partial effects) are not obtainable. This is because the partial effects rely on $\alpha_i$ (fixed effects), which can't be estimated by FE logit.

If you are willing, and I strongly recommend it, please consider Correlated Random Effects (CRE) probit or logit models, where time-invariant unobservable heterogeneity correlated with $X$ is accounted for by the Chamberlain-Mundlak device. The CRE probit (see section 5 of the link) is also known as Chamberlain's random effects probit, and are very popular (certainly more popular in application than FE logit). You can get average partial effects for CRE probit models.


Well-informed citizen can contribute better to country's economy? As the title stated, how can a person from the general public contribute to country's economy by being well-informed?

I understand one of the reason would be tax. If a person understands the importance of paying tax, then the person would be understand his responsibility and pay his tax promptly.

However, are there any other aspects can a informed person contribute to country's economy?

how can a person from the general public contribute to country's
  economy by being well-informed?

Primarily, by promoting policies and pursuing models which the informed person identifies as premised on realistic arguments versus those that are detrimental or make no sense.

A well-informed individual on his or her own is unlikely to outweigh activism or lobbying that pursues adverse interests. Nonetheless, that individual has more elements [than uneducated people] to scrutinize, analyze, and --if need be-- dismantle others' arguments. At the bare minimum, the well-informed individual can reasonably discern what would be most convenient (whether domestically or for the economy as a whole).

  I understand one of the reason would be tax. If a person understands
  the importance of paying tax, then the person would be understand his
  responsibility and pay his tax promptly.

Not necessarily. Being well-informed is not deterministic of the rationale and conclusions at which the person will arrive. The well-informed person could reach a conclusion which is the opposite of the conjecture that he or she will become enthusiastic about taxes.

For instance, the individual could identify (as I personally do) that heavy taxation curtails a person's freedom, significantly complicates his or her financial planning, destroys incentives for productivity/innovation/self-improvement/etc., and even is ineffective toward the purposes of "well-fare" state with which politicians attempt to justify brutal levels of taxation.

Additionally, notions of utility as well as time value of money might still prompt a pro-tax advocate to delay his payment of taxes instead of paying them sooner.


Why do we say the US is in debt to China? Apple import iPhones manufactured by factories in China. Apple has to pay for these Chinese manufacturers. So, Apple should be in debt with Chinese manufacturers. But why do we say the US government is in debt to China?

Because, regardless of private debts of Apple, the US government is in debt to China.

  As of September 2014, foreigners owned \$6.06 trillion of U.S. debt, or approximately 47% of the debt held by the public of \$12.8 trillion and 34% of the total debt of $17.8 trillion. As of 2018, the largest holders were China, Japan, Ireland, and Brazil.

https://en.wikipedia.org/wiki/National_debt_of_the_United_States#Foreign_holdings

See also MAJOR FOREIGN HOLDERS OF TREASURY SECURITIES


Why would males' shorter pension's savings produce more, and not less, income? Source: The European Union: A Beginner's Guide (1 ed. 2012) by Alasdair Blair.

  [p 139:]  Gender has traditionally played a key role in the pricing of insurance policies.
    In 2004 an EU Directive prohibited all discrimination based on
  sex in the access to a supply of goods or services. In principle the
  Directive therefore prohibited the use of gender as a method of
  determining insurance premiums and benefits with regard to contracts that were entered into after 21 December 2007. However,
  the Directive also stated differential pricing could be maintained
  where statistical evidence supported such an approach. Insurance
  companies regarded this as crucial because as women drivers are
  statistically proven to have fewer accidents than male drivers, premiums for female insurance policies have generally been lower. In
  a similar way, because women live longer, men have traditionally

  [p 140:]

  received a higher rate from their pension annuities because their
  life expectancy is lower and as such their pension savings are able
  [1.] to produce  more income over a shorter time [End of 1.].

What have I misunderstood about 1? If male pensioners die earlier,  then their pensions would be shorter, and the principal would generate less interest and fewer payments. So why more income?

I suspect your confusion is merely because you don't know what an annuity is. An annuity pays you a regular stream of income until your death (e.g. $10,000 every year until you die).

So say there's a man and a woman. Each is aged 65 and each has \$100,000 in pension savings. Each person uses their pension savings to buy an annuity.

Suppose the financial institution (usually an insurance company) selling them the annuity estimates that the man's life expectancy is 20 years, while the woman's is 40 years. Then the actuarially-fair payout would be \$5,000 per year for the man and \$2,500 per year for the woman (if we ignore interest rates, discount rates, and similar complications).

Hence, for the same amount spent on the annuity (\$100,000), the man gets a highly yearly income (\$5,000 per year) than the woman (\$2,500 per year). This is what your quoted passage means.

(More on how the annuity works: If the man/woman lives longer than the expected 20/40 years, then good for him/her. He/she keeps receiving the agreed-upon yearly payout even past those 20/40 years.

Conversely, if the man/woman lives shorter than the expected 20/40 years, then too bad for him/her. He/she gets the agreed-upon yearly payout only until he/she dies.)