How are recessions even possible if its impossible for banks to lose money? When a bank issues a loan, the money must end up with another bank. Therefore, the banking system as a whole can never lose money. Only the dumb banks which make loans that arent repaid can ever lose money.
Therefore, is a recession only possible if there is a divide between "smart" and "dumb" banks? Is it the case that, in any recession caused by bad loans, there must always be at least one smart bank somewhere that is profiting and solvent? Since the money has to go somewhere it seems there always has to be at least one bank that is least dumb and ends up with all of the money from others. In fact, the only reason m2 would fall at all is a composition effect whereby the dumb banks are more leveraged and fail.

Recessions are defined as fall in output (usually measured as a consecutive two quarters of negative GDP growth) not loss of money. Output (GDP) might very well fall even when firms are on average more profitable and people earn more money.
For example, in the 2020Q2 the median wage in US actually increased according to U.S. Bureau of Labor Statistics, yet US experiences unprecedented $32.9\%$ GDP decline as reported by Bureau of Economic Analysis.
How can that be? Well it is because the lowest paying people are more likely to get unemployed. In the same way, most of the time you will see the least profitable firms stop producing during recessions not the most profitable ones. Hence even when output of economy declines those who remain employed and economic active might actually earn more so the idea that recession is caused by firms or banks loosing/retaining money per se is simply fundamentally wrong. Its not the loss of money, but loss of output that matters.
Generally speaking recession occur when either aggregate demand or aggregate supply (or both) fall, which in turn leads to loss of output. This is because when aggregate demand falls this means people want to consume less goods and services and hence firms will also supply less of them, and when aggregate supply falls the production is affected directly (figure below shows some graphical examples).

Aggregate demand changes for example in response to changing incomes, wealth, inflation expectation or interest rates and aggregate supply in response to technology shocks, changes in firm costs etc. These are just an examples,  you can find more exhaustive discussion in macro textbooks for example Blanchard et al. Macroeconomics: an European Perspective is an excellent resource.
Most recessions do not even originate within the banking system. Rather banking system can be thought of  as an economic highway through which macroeconomic shocks propagate to the rest of the economy. For example, it is generally accepted that Great Recession was caused by collapse of housing bubble not with banks themselves. Rather the banks played role in propagation of the recession. Furthermore,  banks can also exacerbate recessions due to the role they play in money creation as fall of money supply can make recession worse while expansion of money supply can help alleviate them but exploring this is beyond scope of this answer as your question is about how recessions occur (these issues are discussed in most macroeconomic textbooks you can again have a look at the above mentioned Blanchard's textbook).


How much did it 'cost' slave-owning nations to emancipate the slaves as a percentage of GDP? According to this article in the Guardian the UK Treasury tweeted in 2018:

  "Did you know that in 1833, Britain used £20 million pounds - which amounted to 40% of the UK GDP to buy freedom for all slaves in the Empire. The amount of money borrowed for the Slavery Abolition Act was so large that it wasn't paid off until 2015. This means that every living British citizen paid for ending the slave trade."

To which one angry British citizen, Lexington Wright, tweeted back:

  "So basically my father, and his children and grand-children have been paying taxes to compensate those who enslaved our ancestors and you want me to be proud of this fact? Are you f****** insane???"

The Guardian adds as commentary to this exchange:

  Few people in the 1830s would have seen it this way. Compensation was a mechanism of ending a system that millions of people had come to regard as abhorrent and a national disgrace. The abolitionists agonised over it ... the only people who saw a positive in it were the people who spent three decades campaigning for it and would be the beneficiaries of it - the slave owners.

They go on to add:

  The slave owners not only recieved compensation from the British Taxpayer they won another concession, the euphemistically named 'apprenticeship' system. What this meant was that slaves were forced to work on the fields for another further six years without pay after the abolition of slavery.

The above is the background to the following questions:

Q. The compensation was paid for emancipating slaves in the Empire. How much of this was paid for the top five slave-owning nations? Who are they, and how many slaves were emancipated?

**Q. How much was this worth as a percentage of the top slave-owning nations GDP at that time? And in particular, the USA?

Q. Was any additional monies paid by any of those nations governments as compensation?

Q. How did they come up with the figure of £20 million pounds - was this solely based on the market values of slaves at the time or were other factors taken into consideration?

Q. How would one cost the six years of unpaid work by emancipated slaves and how much did this come to?

There are a number of questions but they are all interlinked and prompted by the initial tweet. I can ask them separately if people wish.

Unfortunately, I'm not sure this will really answer the questions you raised to your satisfaction, but I can at least give some of them a try:

First Note: It wasn't 40% of the GDP of the Empire, but rather the 20 million GBP sum was equivalent to 40% of their annual governmental expenditures (or 5% of the Empire's contemporary GDP) source here.

Regarding your first couple of questions: unfortunately, there's no good, useful answer that I can find. If you were interested in calculating these numbers yourself, it might be worthwhile to look at this online database of payments made under the act. That said, it's important to remember that, as colonies of the British Empire, there might well be many reasons that could make the desired "contemporary GDP amount" difficult to calculate. For example, if there's an individual who owned plantations in Trinidad and the Bahamas, they themselves might live in London. Thus, for all practical matters, the compensation payment the plantation owner would receive doesn't flow to either colony, and shouldn't be considered in those colonies GDPs.

As for "how did they come up with 20 million GBPs?" While I haven't seen a specific answer, I'd imagine much the same way compensation pools are created today- by Parliamentary negotiation, estimation and extrapolation. The number itself is known exactly (that is, we know it's 20 million, and that's not an estimation on the part of the Guardian) since it's inscribed in paragraph XXIV of the act itself (have to scroll down a bit, or search on the page for "million"). That said, there's specific information in paragraph XLV of that act how the sum of 20 million GBP should be divvied up between the different colonies, and that's based on the local "value" of slaves in each colony for the previous eight years.

About additional compensation- no. Remember, there weren't really those "national governments" at the time. They all were, by definition of the act, only territories which were still completely ruled by the British crown. As far as I know, no regional governors supplemented that pool (I'm not even sure if they would have been allowed to, since the authorization for the payments in this case were from the British Parliament, and the vehicle to disperse the payments was a commission set up by the act).

Finally, I'm not exactly sure what you mean by "how would you cost the six years of unpaid labor." Who are you referring to "costing" this? The value of this labor wasn't included in the payments by Parliament. If you were thinking about what the effective cost of this provision was (that is, how much it cost the slaves whose freedom was delayed by those six years) then again, I think it might be an open question.

Again, I'll check back later to see if I can help by refining my answer. Let me know how this aligns with what you were looking for. Also, these additional resources might provide a bit of useful info!
NB: the total number of slaves freed uner this act is roughly 800,000


With inflation so low, why is it so hard to stimulate the economy? Usually the limiting factor in stimulating an economy is how much inflation can a currency handle before the money being spent has a negative effect overall. But now we seem to have the opposite problem of having a lot of leeway, but somehow we are not able to use it to stimulate growth.

It seems like in theory, we could print a bunch of money and do a helicopter drop at the expense of higher inflation, which would help since the US is almost in a deflationary state. Obviously central banks (mostly US but also the rest of the developed world) have better ideas as to what should be done to help the economy, and so I would like to understand what those ideas might be and why they cannot take advantage of the low inflation.

It's not theoretically hard to stimulate the economy. It's just the means to do so are terribly unfashionable. And that means it's politically hard.

We've had two similar depressions in the last couple of centuries: the Long Depression in the second half of the nineteenth century, and the Great Depression in the first half of the twentieth. Each saw significant monetary contraction as a policy response. Which was astonishingly daft.

But after the Great Depression, we also learnt that monetary expansion was insufficient when there's a large debt hangover and demand is suppressed: hence we got the famous analogy of monetary expansion being like pushing on a piece of string.

For many centuries, a once-in-a-human-lifetime jubilee - a cancellation of all personal debts - was seen a way to cure these huge debt hangovers. And they do, broadly speaking, provide a means of cutting the closed circle of monetary expansion which takes place through savings and loans. These jubilees used to happen once every fifty years - and that was, roughly speaking, a human lifetime. A jubilee removes all of the demand-suppression effect that personal debt has: it removes wealth from those with the lowest propensity to spend (thus has a small negative effect on demand), and increases the wealth of those with the highest propensity to spend (which has a much larger positive effect on demand). The net result is a return to increasing demand. In some ways, the bailing out of the banks in 2008 was the very opposite of a jubilee: commercial borrowers (bank bond-holders) were bailed out, and private borrowers were left hanging. Expensive and ineffective. A jubilee cancelled all private debt, but did not touch commercial debt.

The other very unfashionable intervention is fiscal expansion. Keynesian expansions were a successful response to a demand-led depresssion, when the multiplier was higher (as it is now: even Lagarde of the IMF admitted this); but they also got used in subsequent decades in other circumstances too, when the multiplier was lower, and showed less success. And that's when they dropped out of favour, and remain so in many countries now, despite them being part of the right thing to do in the current particular circumstances.


Finding a walrasian demand function If I have a sum utility function like this $U(y) = \sum_{j=1}^J u(y_j)$ where $u$ is convex.

Is there then a way to find walrasian demand for such a function without using calculus or do you need to use lagrange to solve for j,...,J?

Given that the utility maximization problem is:
\begin{eqnarray*} \max_{\{y_j \geq 0 : 1\leq j\leq J\}} && \sum_{j=1}^{J} u(y_j) \\ \text{s.t.} && \sum_{j=1}^{J} p_jy_j \leq m\end{eqnarray*}
where $p_j > 0$ for every $j$, and $m > 0$.

If $u$ is convex and strictly increasing, then in optimum consumer will always spend all his money on the cheapest good. That is, buying $\frac{m}{p_1}$ units of commodity 1 when $p_1 \leq \min_j p_j$ will be utility maximizing for the consumer.

Here is the proof: First of all note that in optimum consumer will never spend less than his income because his utility is increasing. So, we are left with bundles on which he spends all his income. Now any consumption bundle that costs exactly equal to his income can be written as the convex combination of the bundles of the type in which the consumer spends all his money on a single good.

For example, in a two commodity case: Any $(y_1, y_2)$ on the budget line can be written as the convex combination of $\left(\frac{m}{p_1}, 0\right)$ and $\left(0,\frac{m}{p_2}\right)$.

Since, $u$ is convex, and the sum of convex functions is convex, so we also know that the utility $U$ is convex. By convexity of $U$, utility at the convex combination of the bundles of the type in which the consumer spends all his money on a single good is less than or equal to the convex combination of the utilities at those bundles, and that is in turn less than or equal to the maximum of the utilities at those bundles. This shows that the consumer can never lose by changing his choice to the bundle where he buys the cheapest bundle.

For example: In the two commodity case, we can write any bundle any bundle $(y_1, y_2)$ on the budget line as $\lambda\left(\frac{m}{p_1}, 0\right)+(1-\lambda)\left(0,\frac{m}{p_2}\right)$ where $0 \leq \lambda \leq 1$. Therefore, by convexity of $U$,

\begin{eqnarray*} U(y_1, y_2) = U\left(\lambda\left(\frac{m}{p_1}, 0\right)+(1-\lambda)\left(0,\frac{m}{p_2}\right)\right) & \leq & \lambda U\left(\frac{m}{p_1}, 0\right)+(1-\lambda)U\left(0,\frac{m}{p_2}\right) \\ & \leq & \max\left( U\left(\frac{m}{p_1}, 0\right),U\left(0,\frac{m}{p_2}\right)\right)\end{eqnarray*}
This implies that the utility is maximum at either $\left(\frac{m}{p_1}, 0\right)$ or $\left(0,\frac{m}{p_2}\right)$. Given that $U(y_1, y_2) = u(y_1)+u(y_2)$, it follows that it is maximum when one consumes the cheaper one.


Can someone please explain how personal savings and fiscal deficit are related in this context? The lines which i have highlighted is what I want to be explained to me. This is with reference to indian economy

The private sector is households and companies.  The public sector is the state and national governments.  When the author says "private investment" they presumably mean non-financial investment like homes and educations for household members or business processes and software and factories for companies.
The money the private sector wants to save can go into "private investment" or a loan to the government. The more the government wants to borrow the higher the rate of return required by lenders. That rate is the interest rate.
In actuality a household buying a government bond can also be described as making a "private investment" but then it would not make sense to suggest there is "not enough left over for private investment". In order to make a sensible interpretation it is necessary to believe the "private investment" the author intends is non-financial (like a home or factory).


Futures delivery price Can someone tell me how to find the 'delivery price' on a futures exchange? This is the one detail I cannot find on a real futures exchange.

Different futures contracts have different delivery/final settlement machanisms. You need to look up each contract’s specifications.

The U.S. Treasury bond/note futures are an example of a complicated delivery mechanism. There are rules that set the basket of eligible bonds to delivery, as well as the factor relating the futures price to the price paid for the actual bond. That is, the price depends on what bond you deliver.

In the comments, you added more information. The key to understanding futures contracts is that holders get a daily profit/loss each day based on the change in the final settlement price. This means that even though the price for final delivery will likely be different than the price at which you entered the futures contract, you will get a gain/loss from the settlement process that matches the difference. The net result is that you effectively buy or sell at the fixed futures price you entered the position at; you just do not know the sequence of payments that occur as part of the settlement process.

For example, assume that you want to buy a commodity that is traded on a futures exchange. You enter into a long futures position at price of \$1000. If the futures price fell to 900 by the delivery date, you would pay \$900 for the delivery. However, you woukd have lost \$100 as a result of the daily settlement process on your long futures position. Your total cost is \$1000 - where you entered the position. The fact that there are up front gains/losses creates some complexity for fixed income, but that is a second order effect.

If you search around, you should be able to find materials explaining the difference between using futures to hedge versus a forward sale/purchase. That might help clear up your undertanding.


What is an opportunity cost? Mankiw's definition and explanation of opportunity cost here is confusing. Since when have explicit costs become part of opportunity cost?
Here is what the guide says:

  The concept of opportunity cost is one of the most important ideas in
  economics. Consider the question, “How much does it cost to go to
  college for a year?” We couldadd up the direct costs like tuition,
  books, school supplies, etc. These are examples of explicit costs,
  i.e., costs that require a money payment. However, these costs are
  small compared to the value of the time it takes to attend class, do
  homework, etc. The amount that the student could have earned if she had
  worked rather than attended school is theimplicit cost of attending
  college. Implicit costs are costs that do not require a money payment.
  The opportunity cost includes both explicit and implicit costs.

Since explicit costs are out of pocket costs, they are not considered as opportunity cost. Am I missing anything here?

EDIT:
Here is another explanation from Keat et al. Managerial Economics.

  Opportunity Cost Versus Out-of-Pocket
  Cost Previous discussions
  pointed out that opportunity cost is one of the most important and
  useful concepts in economic analysis because it highlights the
  consequences of making choices under conditions of scarcity. We can
  now use this term in a more specific way to help explain the concept
  of relevant cost. Opportunity cost, as you recall, is the amount or
  subjective value that is forgone in choosing one activity over the
  next best alternative. This type of cost can be contrasted with
  “out-of-pocket cost.” On occasion, economists refer to opportunity
  cost as indirect cost or implicit cost, and refer to out-of-pocket
  cost as direct cost or explicit cost.

Keat is mistaken. Most economists agree with Mankiw. The term oppportunity cost includes all costs, including explicit out-of-pocket ones and any other implicit ones.
As Henderson states:

The word “opportunity” in “opportunity cost” is actually redundant. The cost of using something is already the value of the highest-valued alternative use. But as contract lawyers and airplane pilots know, redundancy can be a virtue. In this case, its virtue is to remind us that the cost of using a resource arises from the value of what it could be used for instead.

Here are some definitions from introductory economics textbooks:

The next-best good that is forgone represents the opportunity cost of a decision. (Samuelson & Nordhaus, Economics, 2010, p. 13)
Opportunity cost is the benefit that you might have gained from choosing the next-best alternative. (Colander, Microeconomics, 2017, p. 9)
We refer to this best alternative activity as the opportunity cost. (Acemoğlu, Laibson, & List, Microeconomics, 2016, p. 41)
Economists call that kind of cost—what you must give up in order to get an item you want—the opportunity cost of that item. ... The opportunity cost of an item—what you must give up in order to get it—is its true cost. (Krugman & Wells, Economics, 2015, p. 7).

Of course, as mentioned by Henry, you can define any term as you please. So the question might be, "What is the standard or most commonly used definition among economists?" And the answer is, "Mankiw's and not Keat's."


Econometric model for industrial production I have to build a multivariate regression model as a project for my course in time series analysis. I have decided to model the industrial production index of the USA. I am stuck on the selection of the regressors which can adequately explain the time series of the industrial production. I have already selected some regressors (producer price index, unemployment, interest rates, population, auto and truck sales, inflation, philly fed seasonal adj. index, PMI) but I think I need some more regressors to correctly specify the model.

Can you suggest me some macroeconomic time series I can use as a regressor for the industrial production index?
I need data from 1960-70 to 2016.

Depending on the goal of your analysis, you may want to take a different approach; but, often, more regressors do not necessarily improve a model. Rather than throwing in as many potentially relate variables into your regression, you may want to start with a more theoretical approach to try an determine the underlying supply side and demand side factors that drive industrial output.

It seems to me that the regressors you have included can be broken down in this way:

Supply Side: producer price index and interest rates
Demand Side: population, inflation, interest rates, unemployment, and seasonal adjustment
Other: auto and truck sales and PMI

While I am sure that the "Other" regressors are correlated with your dependent variable, do you really think they are determinants?

That said, if you were to include a lagged term for auto sales you could make the argument that the more cars/trucks bought in the last period would influence the demand for gas and electric in the current period. Additionally, you may have an argument for the PMI, but my point is that you should try and think of the underlying variables driving the supply and demand for industrial production.

Given the above comments, I would say you could add these potential variables:

Supply Side: Government investment in infrastructure, gross fixed capital formation (for the relevant industries), oil prices
Demand Side: Per capita income, manufacturing exports, oil prices

I am sure there have also been many studies like this in the past, and the previous literature is always a good place too look for additional ideas in the modeling process.


Conditions for Radner Equilibrium This is definition of Radner Equilibrium from Microeconomic Theory (Mas-Collel, Whinston and Green - Third Edition).

I'm confused about two conditions:

$\sum_k q_k \cdot z_{ki} \le 0$ (in yellow)

The autors state that this is budget restriction. I wonder how the sum of transactions times the price could be different from zero: if Adam sells a stock for 10 dollars to Ben, the investment positions for Adam and Ben will be, respectively, -10 and 10, summing to $-10+10=0$

$\sum_i z^*_{ki} \le 0$ (in purple)

This is proposed as a feasibility condition. Why?

Condition in yellow: Each consumer $i$ is assumed to have initial wealth endowment $0$.

Condition in purple: For each asset $k$, aggregate demand $\leq$ aggregate supply, which is zero by assumption.

In practical terms, the asset $z_k$ being traded at $t = 0$ are $t = 1$ claims to commodity $k$. It is a derivative contract. In this particular case, it is a futures contract, i.e. the right and obligation to $z_k$ units of commodity $k$.

Assume the exogenous asset returns $r_{sk} = 1$ for all $k$ and $s$.

If you sell a futures contract for $z = 30$ barrels of West Texas Intermediate crude (tons of wheat/thousand pounds of coffee beans/etc.) at futures price $q = \$10$, then you pocket $z \cdot q = \$300$ at $t = 0$. At $t = 1$, you must deliver $30$ barrels of WTI crude to your counterparty. So you must purchase $z = 30$ barrels of WTI at spot price $p = \$20$, say, at a cost of $z \cdot p = \$600$.

Above is the case of a short position. Similarly, if you take a long position, it will cost $\$300$ today but tomorrow you get to have $30$ barrels of WIT, which you get to sell at the spot price $p = \$20$.
This adds $\$600$ to your $t=1$ budget which can then be used to buy consumption good $x_i$'s.

Regardless of long or short position, today's cost is $z \cdot q$ and tomorrow's gain is $z \cdot p$.

If you have zero $t = 0$ wealth, the total value of contracts you buy cannot be more than the total value of contracts you sold. This is condition in yellow.

For each commodity good, if there is an aggregate of, say, $100$ contracts being sold at $t = 0$, there cannot be $101$ being bought.
$$
\mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} \leq 0.
$$
This is condition in purple.

Allowing for $< 0$ implicitly assumes that there are other agents unmodelled that absorbs excess supply. Alternatively, one can insist that the futures market clears, i.e. $$
\mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} = 0.
$$
This would not lead to material change in the economic implications of the model.

Comments

Consumption Smoothing
Under convexity and regularity assumptions on $U$, it is clear from the FOC that agents in this economy trade $t = 1$ claims at $t = 0$ in order to smooth their $t = 1$ consumption across states.

Sequential Trading
In this economy, there is a futures/forward market
at $t = 0$ and a spot market at $t =1$. Agents trade today subject to their budget constraint today, same for tomorrow.

Rational Expectations
However, agents correctly anticipates the $t=1$ spot price at $t = 0$. Equilibrium is self-fulfilling.

Radner vs. Arrow-Debreu
In an Arrow-Debreu equilibrium, agents trade $t=0$ state-contingent claims ("Arrow-Debreu securities"). There is no $t=1$ spot market. Trading only occurs once ex ante. In practical terms, AD securities are digital options, instead of futures contracts.
A priori, agents' choice set of consumption profiles is more restricted in the Radner economy. A futures contract is a un-contingent claim. An important result is the following:

If asset returns ${r_{sk}}$ is complete, then an Arrow-Debreu equilibrium can be implemented by a Radner equilibrium.

This is not surprising. When the market is complete, i.e. there is enough futures contracts to span the set of states, any Arrow-Debreu payoff profile can be engineered from a suitable portfolio of futures contracts.


Who paid the difference on a warrant if i am winnnig I am discovering the mechanism of a warrant but there is still something that i can't understand.

Let's take a simple example a call warrant with a strike of \$10. I buy it when the underlying product is at \$5 and now the underlying is at \$20. If we keep it simple who will give me the \$20-\$10 = \$10 difference.

The bank I suppose. Does the bank hope that on all its warrants it will earn money? Can the warrant put a bank on bankruptcy in extreme case?

A warrant is very similar to a call option (see "Comparison with call options" section here: https://en.wikipedia.org/wiki/Warrant_(finance)#Comparison_with_call_options)

So we can suppose we're dealing with a call option: If $P_t$ is the price of the underlying asset at time t and $K$ is the strike price, then the value of the call option is $V_t=\max(P_t-K,0)$ (if the option has not been exercised by time $t$). This value function indicates that if I own the call option, I will not exercise my right to buy the underlying asset at price $K$ unless the asset is worth more than $K$ at time $t$. By exercising my right at time $t$, I gain $P_t-K$, and the issuer of the option, by selling me an asset worth $P_t$ at a lower price $K$, loses exactly the amount I gain.

Your question seems to be getting at the issuer's intent: Why would you issue a warrant where you would have to pay money on average?

A warrant can be used to:

Make bets, or
Provide monetary compensation without giving up cash flow immediately.

1) As you've suggested, the bank may be hoping to never pay out a warrant. But if everyone else also believes that the value of the warrant is 0, then why would the warrant exist in the first place? Maybe the bank has information that others don't, so some people will buy the warrants from the bank, believing that the warrants have positive value less than the price they paid for them. This would be a use of warrants for betting, both for the bank and the people who purchase the warrants.

2) Companies that have promising future cash flow prospects but have poor short-term cash flow would often like to compensate their workers highly to attract talent, but cannot afford to do so immediately with money (e.g. tech startups). Stock warrants will go up in value with the company's value, so such non-cash compensation is often acceptable as a (partial) substitute for cash salary.


Congruence of GDP as calculated by production and by consumption Here I've read the following (translated by Google)

GDP can be calculated in two ways: firstly, by origin, that is, by
estimating the value of all goods and services produced, and secondly
by the use side, that is, by recording total
expenditure on consumer and capital goods, including external trade
(the difference between the value of imported and exported goods and
services).
These two calculations are always congruent, as consumption
adjusted for net exports corresponds to production.

I wonder about one thing: What about all the products being produced but never sold and used: food, clothes, printed media, to name just a few. They appear only on the one side, not on the other. So how can the two numbers be congruent?
And if they are not congruent: Which of them is finally published e.g. by the German Statistisches Bundesamt?

The GDP expenditure approach is the sum of consumption, private investment, government spending, and the difference between exports and imports. I have not dug into the German Statistisches Bundesamt but unsold goods should be accounted for in the expenditure approach under "change in private inventories" or "inventory investment" (grouped with private investment).


What were "scissors prices" (China, 1950s)? Chan (2018) writes:

  Implementing such a strategy, the state siphoned off resources in the rural sector for capital accumulation in industry through the well-known process of "scissors prices".

What was the well-known process of "scissors prices"?

Under Mao the Chinese government was a planned economy, prices were set by the party, not by the free market. Procurement prices of agricultural produce, what the government payed to farmers, were lower than consumer prices of the same produce, what the government got from selling the produce to the consumers. The difference in prices becomes state revenue. The widening of the gap between the two prices is likened to the opening of a pair of scissors, hence the name.


Do you get economies of scope from horizontal or vertical integration? Economies of scope: make it cheaper to produce a range of products together than produce each one on its own as company can share inputs (e.g marketing and finance) removing cost redundancy

Would this come from horizontal or vertical integration? + example would be great!

Also, is economies of scope the same as synergy?

We'll you've pretty much answered it yourself...
Assuming any integration happens, it'll be horizontal

Vertical integration is buying your customer/ supplier up/down the value chain

Horizontal will be to buy your competitor, and in the general case, its added value and the reason of purchase will be the specific products it creates, not its finance dept. for that matter

Regarding an example, most M&As I can think of right now are of vertical integration
If I remember any I'll add them as comments


What effect would raising reserve requirements while printing money have? It is generally accepted, that printing money will be inflationary, as it increases the money supply without a corresponding real growth of the economy.

At the same time, if the central bank increases reserve requirements (or the official cash rate/bank rate) this decreases the money supply.

So the question is, if a government, perhaps in a time of need, both prints money in order to carry out some project, and also increases reserve requirements/bank rate in order to conteract the inflationary effect, what are the likely follow on economic effects?

During the global financial crisis of 2008-2009, both the US and UK undertook quantitative easing (QE) in conjunction with a policy to fortify banks against risk by increasing reserve requirements.  So there are recent cases which illustrate what happens.

A rise in reserve requirements constrains a bank's ability to lend.  So it seems to counteract the idea of QE.  So why did both the US and UK implement these two seemingly antithetical policies?

Because the two policies also accomplish other objectives.

An increase in capital requirements for banks was thought to be necessary to restore investor confidence in a badly shaken credit system (raising reserves was not done to counteract inflation).  Investors did not have confidence in the quality of assets on bank balance sheets, so requiring banks to retain cash deposits was one way of ensuring that banks had (a) a clear base of assets that were not obfuscated (e.g. subprime); and (b) limitations on how much risk (via lending) they could undertake.  So investors could feel safer about their deposits and the fundamental strength of the banking system.
Quantitative easing was aimed at injecting liquidity into an economy paralyzed by loss of confidence and liquidity, resulting in low spending and investment levels.  In order to offset the negative effects of raising capital reserves, the QE programs needed to be very large, which is why vast amounts of money were pumped into circulation in both countries.

The second part of your question is around follow-on economic effects.  Using the dynamics above you can infer the likely effects.

Raising capital requirements had the effect of exactly that.  It also had the effect of requiring larger volume of QE to counteract the drag on lending (in effect, money velocity or money multipler) cause by the higher reserve requirements.
Had QE not been sufficient to overcome the higher capital  reserves, you would have seen bank balance sheets swell while lending remained low....which is exactly what happened in the early to mid stages of QE while banks shored up their balance sheets before timidly starting to lend at risk again.


Could price floors improve worker safety in the meat cutting industry? Could price floors improve worker safety in the meat cutting industry?
My question is inspired by the Future Perfect podcast https://www.vox.com/future-perfect/21502225/chicken-meatpacking-plant-future-perfect-podcast which discusses the poor worker safety record of the meat cutting industry.
My thoughts are that if a price floor is binding that meat companies might compete on product quality including production practices like worker safety. Also, companies that take worker safety seriously could also enter the market since the price floor could support the higher costs of improved worker safety.

A priori this is difficult to say without any empirical experiment but it is unlikely due to:

Price floors do not necessarily lead to more competition on quality. Theoretical models do show that price floors can (depending on parameters) lead to an increase in quality (Bilotkach, 2012 and sources cited therein), but they can also just result in intensifying competition along different dimensions (such as location, advertising, and other dimensions without necessarily more differentiation of products (see Bhaskar (2003) and Bilotkach (2012) and sources cited therein).
So already an issue is that price floors cannot be guaranteed to lead to higher completion on quality. This would first have to be tested in the specific case.

High-quality products do not necessarily imply better working conditions and high safety standards. Apple products could be argued to be of high quality (although I am an economist, not an engineer), yet apple is getting regularly lambasted for poor working conditions in its factories (see this article from the Guardian).
I was not able to find on google scholar any empirical or theoretical work that would support the idea that price floors on products would lead to better working conditions. Also higher firm profitability is not necessarily associated with better working conditions. Firms can as well just keep working conditions the same and pocket higher profits.

Even if it would work in public economics we are not interested in solutions that just work, but the solutions that achieve goals that we set in an efficient manner. As mentioned by BKay in their excellent comment there are many different ways how to raise work safety, arguably most if not all ideas mentioned in the comment would be more efficient than price floor which would likely have to change with market conditions and be set in a careful manner which might be too difficult to implement.


Does economic theory support the notion that the wealth of the wealthy is based on the poverty of the poor? Almost any discussion about poverty and wealth and income inequality at some point includes arguments based on the premise that the wealth of the wealthy is causally related to the poverty of the poor; more specifically, there often seems to be a tacit agreement that the former causes the latter.

This is the basis for many arguments about distributional justice and in particular the notion that inequality is unjust per se or simply socially inefficient. I am however not interested in the discussion of any ethical questions regarding this issue, as this would motivate primarily opinion-based responses. Instead, I would like to know if there exist any (mathematical) models which support the common assumption that the same economic processes which make the wealthy wealthy also make the poor poor.

Let me preface this answer with a word of caution: your question is a very good and important one but it is also one that depends tremendously on the definitions of the terms it uses. I'm going to attempt to answer it in the most unassuming and non-technical way. You could pose it in technical terms and get a more precise answer.

The wealth of the rich can be attributed to the poverty of the poor. In North Korea, one of the poorest countries in terms of per-capita GDP, there are few rich people and many poor people. It is well-known that almost all of the rich people work either in the highest levels of government or the military. The poor people who are lucky enough to make any income largely do so at the behest of the rich people, and those who refuse are imprisoned or worse. Since almost all economic activity in North Korea is centrally planned via force, it follows that the rich derive their wealth from the suffering of the poor.

However, in societies with more private ownership of production and capital - and less centralization - force is still useful in general but less of a factor in determining who is wealthy and who is poor.

Regardless of our relative lot in life, if you and I freely trade with each other, we only do so when both of us feel like we are "better off" as as a result. So, if you are a wealthy industrialist and I am a poor banana farmer, and I freely decide to trade 100 of my bananas for 100 of your dollars, you necessarily think a banana is worth at least 1 dollar to you, and I necessarily think a banana is worth at most 1 dollar to me.

Excluding outright fraud (misrepresentation either of your dollars or my bananas) we must conclude that each of us either thinks this is a fair trade or that the other guy is a dope. For instance, you may know something about the value of bananas that I don't. Maybe you've just discovered that bananas have special healing powers or that my bananas are particularly good. On the other hand, maybe I know something you don't. Maybe I know that there are about to be many more banana producers in the area or that I can buy two bananas for 1 dollar somewhere else.

Usually, however, it's just a matter of what each party in the trade is able to do with the traded goods that determines what he/she thinks they are worth.

If you get rich from selling my bananas at a higher price, you are not doing so "at my expense" as long as you are not preventing me from doing the same thing. There may be reasons you have the ability to sell bananas at a higher price that have nothing to do with force - maybe you own a distribution system or retail outlets, and I don't - but my choice isn't between selling retail bananas and selling wholesale bananas, but rather selling bananas at the highest possible price vs not selling bananas at all.


Is poverty rightly measured internationally? According to the World Bank, Argentina has 3.63% or people living in poverty. As a subjective assessment of someone living there, this is completely ridiculous. Rates measured by Argentine government and Argentine private institutions set it around 32% to 34%, and they are widely accepted as representative figures of Argentina's today reality.

As far as I understand, the definition used by the World Bank to define a person as poor says someone is poor when he earns less than US\$ 3.1 a day. That makes US\$ 90.30 dollars a month. Now I understand that the living costs are different in different countries, but how could they reach a figure of US\$ 90.30 a month to cover basic needs? How can a person with US\$ 90.30 dollars a month cover basic needs anywhere?

This measure of poverty depends on your definition of basic needs and local prices. Argentina is a good example for how local prices matter. It is widely accepted that official price and inflation statistics published by the Argentinian government are cooked and inflation rates in the last few years are have been substantially bigger than official numbers suggest. I assume that the world bank has to rely on those statistics and it is therefore likely that its  estimates of poverty in Argentina are too low, simply because the price level adjustment is not done with good data.

Alberto Cavallo has a nice paper showing the discrepancies of official statistics with other measures of Inflation in Argentina, see here.

Living on 90$ a month is obviously not nice, but around the world many people do. This measure is intended to document the extent of  extreme poverty in developing countries, i.e. the circumstances where you are literally close to starving. Many more developed countries use a much higher poverty line in their domestic statistics. The world bank as an international body publishes statistics on development that are internationally comparable, and hence uses the same threshold for each country.

The World Bank's extreme poverty line used to be much lower - I believe it started out at 1$ per day around 1990. You can read up on the intention of publishing such a statistic, and how the threshold is chosen in this interesting blog post by a World Bank economist.


Weak and strict dominance Im just seeking for confirmation, is it OK to say that :
"A strictly dominant strategy is a weakly dominant strategy too"?
This would be usefull because for IEWDS we can eliminate strictly dominant strategies first.

Since you mention IEWDS, I presume by "dominant" you actually mean "dominated".
Any strictly dominated strategy would satisfy the condition defining weakly dominated strategies and hence be called such. And yes, strictly dominated strategies can (and should) be eliminated in the process of IEWDS.
The possible typo notwithstanding, any strictly dominant strategy would satisfy the condition defining weakly dominant strategies and hence be called such.


Re-evaluation of Consumer Price Index (C.P.I.) weights How are the weights of the various household items in the Consumer Price Index (C.P.I.) basket decided, i.e., how are they re-evaluated every year? Is it done manually or automatically?

If its done manually, then how can they be sure (i.e., what methods, statistical or mathematical) that the weights represent the general average consumption of that item in the household? Else, if  its done automatically, then what is the strategy (i.e., what sort of algorithms or data structures) used to ensure the accuracy of the weights.

I don't think they are "sure", but the weights are based on the American Consumer Expenditure Survey which has a big sample size, a carefully chosen sampling frame, professionally executed sample gathering, and a large team of experts crafting questions and collating the data.

  How is the CPI market basket determined?

  The CPI market basket is developed from detailed expenditure
  information provided by families and individuals on what they actually
  bought. For the current CPI, this information was collected from the
  Consumer Expenditure Surveys for 2011 and 2012. In each of those
  years, about 7,000 families from around the country provided
  information each quarter on their spending habits in the interview
  survey. To collect information on frequently purchased items, such as
  food and personal care products, another 7,000 families in each of
  these years kept diaries listing everything they bought during a
  2-week period.

  Over the 2 year period, then, expenditure information came from
  approximately 28,000 weekly diaries and 60,000 quarterly interviews
  used to determine the importance, or weight, of the more than 200 item
  categories in the CPI index structure.

Consumer Price Index Frequently Asked Questions (FAQs)

  Consumer expenditure data supplied by the CE Survey are a critical
  component of the CPI, as they are used to estimate weights for the
  CPI’s consumer goods and services classification structure used in the
  calculation of the CPI. In the construction of the CPI, four distinct
  functional uses of CE Survey data are made: (1) to estimate biennial
  expenditure weights, (2) to estimate monthly expenditure weights, (3)
  to calculate the probability that an item’s price will be included in
  the CPI calculations, and (4) to allocate expenditure estimates
  between more-broadly defined expenditure categories from other survey
  sources.

Research highlights of the Consumer Expenditure Survey redesign

Page 6 of BLS Handbook of Methods, April 1997, Chapter 16: Consumer Expenditures and Income has a nice, detailed description of how "The estimation of population quantities of interest, such as the average expenditure per consumer unit on a particular
item category, is achieved through the use of weights." Be sure to check that out too.


What are some applications of Akerlof's 'lemons' in goods and services markets? How can the idea of Akerlof's 'lemons' be applied to goods and services markets (that doesn't include second-hand cars)?

Health insurance. If an insurance company must charge a single premium because they cannot distinguish between low risk and high risk individuals, they will charge a premium based on average risk. As a result the proportion of high risk individuals in the pool of insured individuals increases, whilst the proportion of low risk individuals decreases (healthy individuals are less willing to pay a high price of insurance), which pushes up the average price of premiums, further reduces the proportion of low risk individuals, etc. The market fails.


Getting a market supply curve from firm supply curve PFA the image below.

As there are n firms in the market, and a tax of t moves up the cost curves of an individual firm by t units, shouldn't the cost curve of the market move up by n*t units? If yes, why does the book move the curves up only by t units. If no, could you please explain where my reasoning is going wrong?

Thanks

The important thing to note is that the unit on the vertical axis is dollars per unit.

So if there is a tax of $\$t$ per unit, then the market supply curve will move up by $ \$ t$. (And not by $ \$nt $ as you have suggested.)


Probability of history occurring at time $t$ I am reading LS book "Recursive Macroeconomics Theory".

In this book, as well as in some online lecture notes on the Internet. They often denote, for example, $\pi_t(s^t)$ for probability that history $s^t$ happens at time $t$.

But I wonder, if we are mentioning $s^t$, it implies time is $t$. So, why don't we just make it simple by writing $\pi(s^t)$ instead?

Thank in advance for any help!

You are right, it is the same thing. Actually, later on the book the density over the history $s^t = [s_t, s_{t-1}, \cdots, s_0]$ is written as

$$
\pi(s^t) = \pi(s_t|s_{t-1}) \cdots \pi (s_1|s_0)\pi(s_0) \tag{2.3.1}
$$

where $\pi(s_0)$ denotes the probability of the initial state (or $\pi_0(s_0)$ if you wish), and $\pi(s|s')$ is a transition probability.

Note that Eq. (2.3.1) encapsulates the the probability of obtaining a given history in a Markov process, since ${\rm Prob}(s_t|s_{t-1}, s_{t-2}\cdots,s_0) = {\rm Prob}(s_t|s_{t-1}) = \pi(s_t|s_{t-1})$. This just to show that $\pi_t(s^t)\color{blue}{\equiv}\pi(s^t)$


Hotelling rule and the shadow prices in a peg Hotelling rule states that at equilibrium

$$\frac{\dot p(t)}{p(t)} = i$$

Meaning that rate of change of the price at time $t$ should be equal to $i$, the interest rate. Assume that the government pegs the price at the level $P^{*}$. Assume further that the shadow price at time t is given by: $Q_t =( \sigma i S_t)^\frac{-1}{\sigma}$

Show that if $\dot q(t)= i$  then $Q_t = P^{*}$

My questions:

(1) what exactly is $\dot q(t)$ or $\dot p(t)$. Is it the derivative? I couldn't make sense out of it.

(2) Is the hoteling condition given in logarithms?

(3) And what method to go with after understanding (1)-(2)

EDIT:

Yes, this is a homework question. However I never asked you to solve the question. I asked what kind of a method would be used in this case. Since the problem doesn't make sense.

$$q(t) = log (Q_t) = \frac{-1}{\sigma} log(\sigma i S_t) \\ \frac{dq(t)}{dt}=\dot q(t) = \frac{- \sigma i S_t}{\sigma^2 i} = \frac{- S_t}{\sigma }$$

So condition on $q(t)$ implies that $S_t = - \sigma i$. Which I believe there is a problem since $S_t <0$ Anyway :

since $S_t <0$ at this specified time $t$ all the resources have varied off.

If I ended up finding $S_t = 0$ then I would conclude that at this time $t$ all the speculators bought up the resource and since the resource was sold at the price $P^*$ somehow this would imply $Q_t = P^*$

But $S_t <0$ is the result we get. Which simply says that all the goods were bought plus government bought some goods. I think that implies a price $q(t)>P^*$

Partial Answer
I can give you an answer for parts (1) and (2). As for (3), I can't give an answer unless you show some work as per our site policy on homework questions.

(1)These variables are interpreted as the change in $q$ or $p$ over time. Mathematically this is the derivative, giving us the results:
$$\dot q(t)=\frac{dq(t)}{dt}$$ and $$\dot p(t)=\frac{dp(t)}{dt}$$
(2) Hotelling's rule is not given in logarithms, as explained in as the notation is explained for (1). if it was it would be indicated by the notation of $\hat p(t)$. Written concisely as:
$$\hat p(t)=\ln(p(t))$$


Why does a strong dollar make imported goods cheaper? I'm having a hard time visualizing why a strong dollar would make an imported good cheaper?  If I buy an ounce of gold in the US for \$1,353.70 (todays rate) and I buy an ounce of gold in Europe, would it not cost me 1,092.81 Euros at todays exchange rate of \$1.24 to 1 Euro?  Why would I expect to get to get more than an ounce of gold if I spend \$1,353.70 in Europe?

Consider that you want to purchase a product from a EU country, pricing their products in Euros.

Let’s say that you want to buy a product that costs 100 euro. No matter what the relationship is between USD/EUR, the product will still be 100 EUR.

Let’s say the USD/EUR is 1.24 as in your example - that means for every 1.24 USD you have, you get 1 EUR. If you want to buy the product you will then need 124 USD to buy a 100 EUR product.

Now let’s say that the USD strengthens compared to EUR and the new exchange rate is 1.15 instead of 1.24. Using same logic as above, you now need 115 USD to purchase that same product, giving you a discount of 9 USD just because of the new exchange rate.

Remember the product is still 100 EUR - but you can now buy more euros for your dollar than you could before.


Decomposing Changes in ROA into Changes in Operating Margin and Asset Turnover via Total Factor Productivity (TFP) I am currently reading a paper on the influence of hedge fund activism on plant-level productivity. In the appendix the author describes how he decomposes changes in ROA into changes in operating margin and asset turn-over. I am trying to redo the calculation explained by the author but am unable to do so.

The author is calculating the change in operating margin via a increase in TFP using the following formula:

Operating Margin = 1-1/(TFP x price change ratio)

The author calculates 26,6 percentage as the new operating margin but I cannot replicate the calculation. Any help is appreciated.

Here the part in question:

  In Appendix B, we link formally the magnitude of the change in ROA to
  the change in raw TFP from years  tt  to  t+3t+3 . In particular, we
  use a modified version of the decomposition in Bosch-Badia (2010), in
  which ROA is decomposed into TFP, input and output price changes, and
  asset turnover. Using the “DuPont decomposition” of ROA, we obtain the
  following relation:

  (A2) ROA=Operating margin × Asset turnover,ROA=Operating margin ×
  Asset turnover,  where ROA is the ratio of earnings before interests
  and taxes (“operating profits”) to lagged total assets, operating
  margin is the ratio of operating profits to concurrent sales, and
  asset turnover is the ratio of sales to lagged assets and, as
  Bosch-Badia (2010) shows, operating margin = 1–1/(TFP × price change
  ratio). The price change ratio is the change in output price divided
  by the change in input price. All price changes are relative to the
  benchmark year (i.e., year  tt ). In addition, we further make the
  following two assumptions: (i) The baseline operating margin is 24.7%
  (see Table 2, Column (1)), and (ii) the price change ratio is equal to
  one (i.e., input and output prices change by the same magnitude). With
  these assumptions we can link the change in ROA to the changes in TFP
  and in asset turnover. First, we estimate the change in TFP using the
  specification in Table 4, Column (4). Specifically, we narrow the
  estimation to only manufacturing firms based on Compustat SIC codes
  and find average productivity gains of 2.6% from years  tt  to  t+3t+3
  for plants owned by manufacturing target firms. Second, given the
  baseline operating margin of 24.7%, the increase in TFP of 2.6%
  translates into an expansion in operating margin by 1.9 percentage
  points to 26.6%. Third, the magnitude of the change in ROA also
  depends on the change in firm-level asset turnover, which is driven by
  reductions in capital at the plants that are not sold, and by
  divestitures and/or closures of plants. Using Compustat data, we find
  that for manufacturing target firms asset turnover increases from 1.07
  to 1.20 on average from years  tt  to  t+3t+3 . Taking the two changes
  together, the implied ROA increases by 5.5 percentage points from
  26.4% in year  tt (= 24.7% × 1.07) to 31.9% in year  t+3t+3  (= 26.6% × 1.20).

And here a direct link to the paper.

Go to Appendix B.

The formula for OM is:

$$ OM = 1 - \frac{1}{TFP\times PCR}$$

In the baseline year, you have that Operative Margin (OM) = 0.247. Further, by assumption, price change ratio (PCR) = 1. Then:

$$TFP = \frac{1}{1-OM} = \frac{1}{1-0.247}=1.328$$

Then, you are told that the TFP changed by 2.6%. Therefore, the new value for TFP is 1.3625 (equivalent to 1.328 $\times$ 1.026). With this new TFP, and assuming that PCR = 1, you can find the new value for OM:

$$ OM = 1 - \frac{1}{1.3625\times 1}$$
$$ OM = 1 - 0.734$$
$$ OM = 0.266$$

There you get the 26.6% result.


Question about the Ellsberg Paradox in Expected Utility Theory The von Neumann-Morgenstern theorem states that, assuming a person's preferences under risk satisfy certain rationality axioms, then there exists a utility function u, the von Neumann utility function, such that the person will tend to maximize the expected value of u.  For this reason, the hypothesis that people satisfy the von Neumann-Morgenstern rationality axioms is known as expected utility theory.  Now one of the major challenges for expected utility theory is the Ellsberg Paradox.  It goes as follows.

Suppose you have an urn which has a total of 90 balls, 30 of which are red and each of the other 60 balls is either black or yellow.  And suppose a ball is drawn at random from the urn.  Then would you rather have lottery A, where you get 100 dollars if a red ball is drawn, or lottery B, where you get 100 dollars if a black ball is drawn?  Most people would prefer lottery A.  And would you rather have lottery C, where you get 100 dollars if you draw a red or yellow ball, or lottery D, where you get 100 dollars if you draw a black or yellow ball?  Most people would prefer lottery D.  But the thing is, preferring both lottery A to lottery B and lottery D to lottery C is inconsistent with expected utility theory; see this Wikipedia article for the proof.

I'd like to understand this logic a little better.  Consider a new urn that just has 60 balls, and each ball is either black or yellow.  Then would you rather have lottery A', where you get 30 dollars guaranteed, or lottery B', where you get a dollar for every black ball in the urn?  I think most people would prefer lottery A'.  And would you rather have lottery C', where you get 30 dollars plus a dollar for every yellow ball in the urn, or lottery D', where you get 60 dollars guaranteed?  I think most people would prefer lottery D'.

So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'?  I think it's consistent with expected utility theory.  Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars?  That would transform my example into the example in Ellsburg's paradox.  So where's the flaw in my reasoning?

The short answer seems to be yes your example violates expected utility... It mostly seems to me like a simple transformation of the first example you gave (but you got rid of the red balls).

As mentioned in other answers expected utility is not equipped to handle uncertainty because it deals with taking expectations and expectations cannot be computed when you don't know probabilities. For that reason, I think it is relevant to my answer to specify the distinction between what risk and uncertainty are.

Risk: This deals with facing a set of probabilities over outcomes in which the agent knows/understands the possible outcomes that he is facing
Uncertainty: This deals with an agent facing an unknown set of probabilities over possible outcomes

In a previous version of my answer, I wasn't careful in how I thought about and explained these ideas and I have clarified a few ideas in my head since then which hopefully translates to a clarified answer. Now to answer your question:

We have an urn with 60 balls. We are uncertain about the number of black balls and yellow balls in this urn. Imagine we present this urn with the above specified lotteries to an individual (with a monotonic utility function $u$) and ask them to choose between $L_{A'}$ and $L_{B'}$. Without loss of generality, imagine they choose to take $L_{A'}$.

This tells us by revealed preference $L_{A'} \succ L_{B'}$ which then implies

$$u(30) > u(B)$$

where $B$ is the number of black balls that are believed to be in the urn (no probabilities here, this is some number the decision maker arrived at and at this point we don't know how). This implies

$$B < 30$$

Now imagine that you give the agent the choice between $L_{C'}$ and $L_{D'}$ and the agent chooses to face lottery $D'$. Then

$$u(60) > u(30 + (60-B))$$

where $(60-B)$ is the number of yellow balls that the agent must think are in the urn. This implies

$$60 > 30 + (60-B) \Rightarrow B > 30$$

This means that an agent cannot prefer $L_{A'}$ to $L_{B'}$ and prefer $L_{C'}$ to $L_{D'}$ because that would imply $B > 30$ and $B < 30$. It is difficult to talk about where $B$ comes from because it is a hard concept to think about when we are used to working in expected value terms, this agent cannot compute any type of expectation over the sets of values that he thinks $B$ lies in. As an example of this, let me use a max-min approach.

Imagine that when he is presented the urn and lotteries that he is told that the number of black balls is either 15 or 45, but the agent doesn't know with what probability it is 15 and what probability it is 45.

Lottery $A'$ will give him $u(30)$ utility units for sure
Lottery $B'$ will give him either $u(15)$ utility units or $u(45)$ utility units.
Lottery $C'$ will give him either $u(75)$ utility units or $u(45)$ utility units.
Lottery $D'$ will give him $u(60)$

The agent is worried about uncertainty (an interpretation of the max-min is that he thinks it is a magic urn that tries its best to pay the least amount as possible, so it will choose the value that leaves him worse off if given a choice). Then using this in his decision process when comparing $L_{A'}$ and $L_{B'}$ he thinks these lotteries will pay the following amounts

$L_{A'}$ will pay $u(30)$
$L_{B'}$ will pay $\min(u(15), u(45)) = u(15)$

thus he chooses lottery $A'$. Now consider $L_{C'}$ and $L_{D'}$. He thinks these lotteries will pay

$L_{C'}$ will pay $\min(u(45), u(75)) = u(45)$
$L_{D'}$ will pay $u(60)$

thus he would choose lottery $D'$. This is one example of how uncertainty can be treated in a problem like this, but by no means is the only.

Note: Earlier I thought had a piece of my answer that talked about priors over the value of $B$, but this wasn't carefully thought through. You can have multiple values that you might imagine $B$ takes, but as soon as you assign a probability distribution to any of these values then you have left the realm of uncertainty and moved to the realm of risk.


Consequences of a particular loss function So I was reading about the consequences of various loss functions on what regressions result in, i.e. the L1 norm giving conditional median estimation and L2 giving conditional mean, etc.

What happens if you minimize the sum of $\left(\frac{\hat{y}}{y}-1 \right)^2$ over all observations? Provided no y != 0, I get Monte Carlo results that are very close to standard OLS results.

I SHOULD be doing my time series homework but instead I've been tinkering around.

Not a super serious question, just curiosity :)

Is it the case that since you can rearrange this expression to $\Sigma (\frac{1}{y^2})(\hat{y}-y)^2$, that you are really weighting the standard L2 norm with  weights that are higher for lower absolute values of y?


Osborne, Nash equilibria and the correctness of beliefs In Osborne's An Introduction to Game Theory Nash equilibrium is described as follows (p. 21–22):

  First, each player chooses her action according to the model of
  rational choice, given her beliefs about the other players' actions.
  Second, every player's belief about the other players' actions is
  correct.

It seems to me that this definition is not completely equivalent to the usual definition of the Nash equilibrium as a strategy profile where each player's strategy is a best response to the strategies of the others.

The usual definition says nothing about beliefs and therefore allows for the possibility that beliefs might be incorrect.

To take a trivial possibility, consider the Prisoner's Dilemma. Suppose each player believes that the other player will not confess. Since confessing is a dominant strategy each player would still confess. So the actions constitute a Nash equilibrium even though the players' beliefs are completely the opposite of the actual equilibrium actions.

Am I right in this understanding that Osborne's definition characterizes something other than Nash's equilibrium?

Introducing the language of beliefs here is slightly strange, given that beliefs do have a very specific meaning in other parts of game theory.

Indeed, Osborne's description is reminiscent of a Bayes Nash Equilibrium. We could introduce the notion of beliefs into the normal form of a complete information game as follows: suppose that with probability $a_i$ each player, $i$, is a "strategic" type who will play according to (Nash) equilibrium, and with probability $1-a_i$ he will select some strategy uniformly at random (because, say, he is indifferent across all actions). We thus have a Bayesian game where thinking about beliefs is more natural.

The Bayes Nash solution concept then says that $i$'s strategy must be optimal given the expected play induced by the other players' strategies and the beliefs over their types implied by $\{a_j\}_{j\neq i}$. If we look at the limit as $a_i\rightarrow 1$ for all $i$ then the Bayes Nash equilibrium of this game will coincide with the solution concept described by Osborne.

I guess the reason Osborne wrote it like this is a pedagogical one, given that this is an introductory text. When we introduce students to static games, we tell them that player $i$ best responds to the actions of the other players. Students naturally want to know "how can they respond to a strategy chosen simultaneously without knowing what that strategy will be?" This is, in many senses, a philosophical question. Common answers are

If the game is one that is played often then (putting aside issues of
other outcomes that can be sustained in repeated games) we can think
of Nash as being an equilibrium in the sense that if we converge
there we can develop a norm whereby people continue to play that
equilibrium indefinitely (and expect others to do the same).
If the game is really one-shot then we usually invoke the idea that players are going to try to predict what others will do—and our equilibrium notion embeds the idea that these predictions must be correct.

It seems that the predictions in the second point correspond to the "beliefs" invoked by Osborne. However, it is important to stress that these predictions/"beliefs", are merely an informal/intuitive tool for helping us to conceptualise what is going on in an equilibrium and are not part of the definition of such an equilibrium. The concept of Nash equilibrium itself is completely agnostic on the notion of beliefs (as you note in a comment, it is defined only over actions), which is why, when Osborne goes on to formally define Nash equilibrium, he does so without invoking the idea of beliefs at all.


What are main methods for econometrics of Macroeconomics? I have a background in Finance and have become pretty interested in Macroeconomics as a graduate student. However the time is limited and I want to focus on econometrics part of Macroeconomics. In that case, could someone please give me tell me what are the main econometrics methods I should learn in Macroeconomics？In Finance we focus on models like ARMA and GARCH, how about Macroeconomics? I've learned VAR before. I also heard about the simultaneous equations, are those for Macro? What are methods that econometricians came up to handle Lucas's Critique? What might be a good book to learn Macro econometrics?

Thanks in advance!

There are many different econometric methods used in macroeconometrics, but the basic ones are advanced versions of least squares methods and Generalized Method of Moments (GMM). Structural VARs, ARCH/GARCH models and several others are widely used as well.

Since you're already familiar with econometric methods, you might try glancing through Fair's "Macroeconometric Modeling" (http://fairmodel.econ.yale.edu/mmm2/mm.pdf). You don't necessarily have to read it completely, but I found it a really useful document in understanding how to build and use macroeconometric models.

Based on your background and interest, I'd recommend Favero's "Applied Macroeconometrics" and De Jong's and Chetan's "Structural Macroeconometrics" as general books on the subject.


Why does the definition of MRS follow from the implicit function theorem? TRAIN OF THOUGHT 1:

From what I understand, $MRS$ is calculated as

$$dU = U_x dx +  U_y dy =0$$
which by rearrangement yields
$$\frac{dy}{dx}= -\frac{U_x}{U_y}$$

So suppose I have $$U(x,y) = \ln x +\ln y$$ Then
$$ \frac{dy}{dx}= -\frac{1/x}{1/y} = -\frac{y}{x}$$

Okay. So I have a function $y$ in terms of $x$.

TRAIN OF THOUGHT 2:

Now consider my $U(x,y)$ again. Let $$\mathbf{a} = \begin{bmatrix}
1\\
1
\end{bmatrix}$$
and $$U(\mathbf{a})=0$$
We have $$DU(x,y) = \begin{bmatrix}
\frac{1}{x} & \frac{1}{y}
\end{bmatrix} $$
and $$\frac{\partial U}{\partial y} (\mathbf{a}) = \begin{bmatrix}\frac{1}{y}\end{bmatrix}= 1$$ which is nonsingular since $\det(1) = 1$ and so by the Implicit Function Theorem, $$U = 0$$ defines $y$ implicitly as a function of $x$ in a neighborhood of $\mathbf{a}$.

My Question:

How are these two trains of thought connected? The first is stated in terms of differentials. But the second is not. So I am confused why the definition of $MRS$ follows from the implicit function theorem.

It's actually pretty straight forward. The implicit function theorem for two variables is given as follows (as long as some regularity conditions hold):

For $F(x, y) = 0$,

$
\frac{dy}{dx} = -\frac{\partial F / \partial x}{\partial F / \partial y}
$

In the case of MRS, we want the marginal change in $x$ associated with a marginal change in $y$ required to maintain a certain level of utility, $c$, such as (conveniently) $c=0$. So, starting with

$U = U(x, y) = 0$,

we have

$\frac{dy}{dx} = -\frac{U_x}{U_y}$

Note that $c=0$ is just a simplification for exposition. For a general $c$, you can just subtract it from either side of the equation and you get the same result since $c$ disappears in the derivative.


Is there an algorithm to find as many NE as possible in this generalized second price auction? Suppose three advertisers, I, II, III are participating in an auction for three positions for online advertising, top, middle and bottom. Assume that click per second  for a position is not affected by who poses an advertisement there. For top, the click through rate is 3 click per second, for middle, it's 2, for bottom, it's 1. But advertisers have private value per click. For I, it's \$16 per click. For II, it's \$15. For III, it's \$14. The one who submits the highest wins the top position and pays the second highest bid, second highest bidder got the middle position and pays the third highest bid. The third highest bidder end up with bottom position, but pays zero. If there's a tie, then choose the winner ramdomly by a fair dice. The strategy space for each bidder is $\mathbb{R}_{+}$.

There're a lot of NE for this game. For example, I,II,III simultaneously submit bids of, \$7 per click, \$9 per click, \$11 per click respectively. Their payoff will be $1 \times (16-0)$ dollars, $2\times (15-7)$ dollars, and $3 \times (14-9)$ dollars respectively.

$$
\begin{array}{c|l|c|r}
\hline
 &(0,7) & (7,9) & (9,11) & (11, +\infty)\\
\hline
\text{player I}& \color{blue}{16}  &  \color{blue}{16}  & 2(16-9)=14& 3(16-11)=15 \\
\hline
\text{player II}& 15 & 2(15-7)=\color{blue}{16} & 2(15-7)=\color{blue}{16}& 3(15-11)=12 \\
\hline
\text{player III}& 14 & 2(14-7)=14 & 3(14-9)= \color{blue}{15} & 3(14-9)= \color{blue}{15}\\
\hline
\end{array}
$$

How to find other NE?

Added: I want to study this numerical example, because I want to understand better the motivation of this paper.

This is a pretty general algorithm, you can probably tailor a better one to your specific problem.

If you stick to this discreet strategy space it seems to me you would have to find the equilibria via brute force. Basically you would look at all bid profiles $(x_1,x_2,x_3)$ and check whether it is an equilibrium. To do this, you would have to check if player 1 would be better off by making another bid. Let us denote the payoff of player 1 by $\pi_1$. This is a function of the bids. Let's assume $x_2 > x_3 + 1$. Then you would have to check
\begin{eqnarray*}
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_2+1,x_2,x_3) \\
\\
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_2-1,x_2,x_3) \\
\\
\pi_1(x_1,x_2,x_3) & \geq & \pi_1(x_3-1,x_2,x_3) \\
\\
\end{eqnarray*}
If all these and the corresponding inequialites for other players hold then you have found an equilibrium.

If $x_2 = x_3 + 1$ you need to take the tie breaking rule into consideration.

You could limit the number of profiles $(x_1,x_2,x_3)$ you check by reasoning about the reservation prices. (E.g. for any N.E. where someone bids higher than their reservation price you have an N.E. where they bid just that.)


What is the difference between a bank's Treasury department and ALCO? I know that the treasury manages and controls the bank's capital. While the Asset-Liability Committee deals with the assets and liabilities in order to stay solvent, right?
What are the similarities and differences between the two?

Someone could write a book about this question which still wouldn't cover all considerations. However from a high-level perspective, I would say that TR handles a lot of cross-functionialities with a main focus on risk mitigation, like especially

liquidity risk
FX risk
counterparty risk
interest rate risk

and most of the times TR also covers the pricing for financial assets (loans, bonds, tender, etc.) on a national bank / central bank level.
Whereas ALCO focuses more on structuring assets & liabilities, ensuring the fulfillment of regulatory figures like core capital ratio, etc.
So they earlier or later will have a lot if interfaces but still their very own tasks and responsibilities.
Side Note: I work in treasury myself (corporate side) and used to work for banks close to 10 years as well


Effectiveness of recycling (consumer PET) plastic Is recycling plastic, all things considered, efficient as compared to disposing of them and making new plastic?

I'm looking in efficiency rates in free market environment and also full process of production and disposal. No moral agenda, no subsidies, simply, is it worth to recycle plastic.

It is not easy task to find all the costs involved and it surely depends on place and politics so I understand some assumptions have to be set. I understand that keeping our backyard clean is also important so when disposal is considered it should be done in proper manner. No enforcement should be speculated (such as enforced sorting by the public), although voluntary action may be considered (motivated public).

The costs of production option of plastic should involve:
production (costs), introducing to the market (gain), disposal (costs)

The costs of recycling option of plastic should involve:
collecting (costs), processing (costs), reintroducing to the market (gain)

Anyone help is appreciated. It's not any project, just curiosity of my own. If anyone can at least point me to good resources and outline how to calculate everything objectively, I would much appreciate that.

One document that deals with the topic is here:
http://cdn.intechopen.com/pdfs-wm/37109.pdf

Although it doesn't give clear answer (hard numbers).

Update: What I seek here mostly is proper methodology of objective economic calculation of this very matter.
1. What are the partial processes to consider - the major contributor of costs/profits.
2. How to determine the costs/profits of each part.
3. Where to find objective data or how to get it.

I should have probably attempted to try something on my own, and I intend to. But once I start thinking about it I find that I either don't have objective output or that the calculation would require too much detail.
Example: landfill cost according to the EU document in comment, p.64 ( http://ec.europa.eu/environment/waste/studies/pdf/eucostwaste.pdf ) the prices range from 6 EUR/ton in Poland to 164 EUR/ton in Netherlands. Now, is the difference due to higher labor costs or due to taxes and directives?

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

  How to determine the costs/profits of each part.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Estimating CES utility (not production) function parameters The CES utility function has the form
\begin{equation}
u(x_1,\dots,x_n)=\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho},
\end{equation}
where $\alpha_i$ is the consumption share parameter and $\sigma=\frac{1}{1-\rho}$ is the elasticity of substitution.

I'm interested in estimating the $\alpha_i$'s and $\sigma$. Since we do not directly observe utility levels, I think we can instead use data to estimate the Marshallian demand functions implied by the utility function:
\begin{equation}
  x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n.
\end{equation}
Clearly, the function is highly nonlinear in the parameters of interest. So a nonlinear estimation procedure would be required.

Say we have data consumption $x_i$, prices $p_i$ and income $M$. How do we go about estimating the parameters $\alpha_i$'s and $\sigma$?

I can find a lot of papers on estimating the CES production function, but I don't find them helpful since I don't observe the dependent variable ($u$). Given that the CES utility is a common functional form used in the literature, I suspect its estimation must be somewhere in the literature.

I managed to find the nonlinear least squares estimation in Stata, but I don't know how susceptible such a procedure is to misspecification and how much confidence one should put on the statistics (e.g. t-stat or p-value) produced by the program.

Any pointer to the papers/textbook chapters on the topic, as well as statistical packages that implement the estimation procedure will be greatly appreciated.

It may be interesting to exploit the homothetic separability of the CES utility function in $x$. It implies that
$$\frac{x_i}{x_j} = \left( \frac{\alpha_i}{\alpha_j}\frac{p_j}{p_i} \right)^\sigma $$
and after $log$-transformation:
$$\ln(x_i) - \ln(x_j) = \beta_{ij} + \sigma (\ln(p_j) - \ln(p_i)). $$
After adding a random term, this specification could be used to estimate the $\beta_{ij} \equiv \sigma (\ln(\alpha_i)-\ln(\alpha_j))$ and $\sigma$ parameters by OLS and in a second step identifying the $\alpha_i$ by minimum distance:
$$ \widehat{\alpha} = \arg \min_\alpha \Big(\widehat{\beta}-\widehat{\sigma}(\ln(\alpha)-\ln(P\alpha)) \Big)'\Omega^{-1} \Big( \widehat{\beta}-\widehat{\sigma}(\ln(\alpha)-\ln(P\alpha)) \Big), $$
where $P$ represents the adequate permutation matrix.
Some references estimating parameters of CES preference parameters include Diewert and Feenstra (2017) or Redding and Weinstein (2020), with a different approach, however, based on the unit expenditure function.
Most empirical contributions reject the validity of homothetic utility functions (like the CES). There are some propositions on how to built nonhomothetic CES production functions that could easily extended to utility functions, see Shimomura (1999) and the references therein. A simple extension of the above specification is:
$$\ln(x_i) - \ln(x_j) = \beta_{ij} + \sigma (\ln(p_j) - \ln(p_i)) + \gamma_{ij} M/p + \varepsilon, $$
where $p$ denotes a consumer and time specific aggregate price index (over all commodities). Notation $ij$ stands for commodities $i$ and $j$ and not for the observations (I skipped the $n,t$ subscripts for clarity).
This relationship is compatible with nonhomothetic preferences. The homothetic case is obtained for $\gamma_{ij}=0$ which can be tested.
Regarding the computer software... I fully switched to R few years ago, it would "easily" allow to code the minimum distance estimator, and the link between the utility and demand functions.
References:
Diewert, Erwin and Robert Feenstra (2017), “Estimating the Benefits and Costs of New and Disappearing Products,” mimeo, University of California at Davis.
Redding Stephen J and David E Weinstein (2020), "Measuring Aggregate Price Indices with Taste Shocks: Theory and Evidence for CES Preferences," The Quarterly Journal of Economics, 135, 503-560.
Shimomura, K., 1999, "A simple proof of the Sato proposition on non-homothetic CES functions," Economic Theory, 14, 501–503.


Convex Preference but Convex Utility Can preference be convex when utility is not a concave function (e.g. $U=x_1^2 + x_2^2$)?

It's well known that a convex preference implies quasiconcave utility functions. Since quasiconcavity need not imply concavity, it's easy to find examples of a non-concave utility function representing a convex preference.
For example: $u(x,y)=(x+y)^3$. The preference this function represents is convex (though not strictly so), as can be seen from its linear indifference curves. The function is quasiconcave, as evidenced by the convex upper contour sets. Lastly, the function is not concave, as betrayed by the exponent.


Predictive Game Theory I was reading this neat piece by Fudenberg http://fudenberg.fas.harvard.edu/predictive%20game%20theory.pdf on Predictive Game Theory. He argues correctly that most traditional work in Game Theory is not suited for prediction in real world set-ups.

Do you know any work that has advanced the theory or empirics of  predictive game theory?

I am aware of some recent work of partial identification for games but I am more interested in learning in games and off-equilibrium dynamics.

Actually, Section 3 of the paper you linked is probably one of the most comprehensive lists you could find of the movers and the shakers with regard to predictive game theory. I know that many of them have both published and working papers addressing these issues. Fudenberg even addresses those specific two topics. A few selections from the paper:

Ignacio Esponda, Philippe Jehiel, and David K. Levine are leaders in studying adaptive processes in extensive form games, and the sorts of non-Nash equilibrium outcomes that can persist even when players have a lot of experience with the game.
Michel Benaïm, Josef Hofbauer, William Sandholm, and Sylvain Sorin are making important advances in the mathematics of dynamical systems and applying them to non-equilibrium dynamics.
Chaim Fershtman and Ariel Pakes are developing estimation methods for field data that allow for incorrect off-path beliefs.
Jeff Shamma is a pioneer in bringing techniques from the feedback-control literature to the study of learning in games.

I would recommend going through these authors' websites and looking for papers that fit.


How to model spendthrift behavior First, I'll introduce the observation and then ask the question.

I have a brother-in-law who has a quirk: given the same product and two prices for it in different stores, he's willing to pay the greater one. For instance, he preferred window panes cut from a more expensive store, even if they are the exact same thing. He also goes to expensive restaurants and order basic meals (say, beef with salad) so there's very little extra they can offer for a steep price increase. Trust me, I can continue providing examples until you convince yourself that he just like to pay more, given the opportunity.

I wonder how would one model such behavior. The easiest way is to give up and say he's irrational (his choices can be easily shown to be intransitive) and be thankful that he's a minority. I'd like to explore other alternatives:

There are prices in the utility function and $\frac{\partial U}{\partial p} \gt 0$ . This solves the problem but it introduces more issues. On the one hand, the price doesn't belong to a basket. On the other hand, he doesn't just spend all of his income to buy just an extremely overpriced glass pane, so they must be introduced carefully.
The basket has a "perceived as expensive" variable that accounts for the behavior. How would such variable differ from being just the price is intriguing...

Can you think of other alternatives, superior to these? Maybe ones that involve consumer aggregation?

It could simply be conspicuous consumption, a case of signaling. He does not buy the more expensive item for quality's sake, but to show that he has the income to shop there. In this model you could tweak his utilities in such a way that he would not spend all his money on needless expenses, but he would spend some of it.


Are business cycle fluctuations usually studied in a (New Keynesian) DSGE model, or can they also be studied in a growth model? I wonder if business cycle fluctuations are usually studied in a (New Keynesian) DSGE model, or can they also be studied in a growth model?

Business cycle fluctuations can be studied in many ways including also growth models. An example of using growth model to study business cycle would be this 1995 paper by Cho & Cooley. This being said growth models are not as common for examining business cycles as for example DSGE models so saying they are usually used for that purpose is correct. To my best knowledge no statistics on the relative use of this or that model for explaining business cycles exist but anecdotally, speaking from attending macro conferences, DSGE or various non-stochastic (when we talk about pure theory) New Keynesian or to lesser extent real business cycle models remain popular.


The relationship between indirect utility and expenditure functions I am trying to understand the fact that $e(p, v(p,y)) = y$. There is a proof in the text Advanced Microeconomic Theory (Jehle and Reny) that states the following:

  Because $u(·)$ is strictly increasing on $R_n^+$, it attains a minimum at $x = 0$, but does not attain a maximum. Moreover, because $u(·)$ is continuous, the set $U$ of attainable utility
  numbers must be an interval. Consequently, $U = [u(0), u^b)]$ for $u^b > u(0)$, and where $u^b$ may be either finite or $+∞$.

  To prove, fix $(p, y) ∈ R^{++}_n × R^+$. We know $e(p, v(p, y)) ≤ y$. We would like to show in fact that equality must hold. So suppose not, that is, suppose $e(p, u)<y$, where $u = v(p, y)$. Note that by definition of $v(·), u ∈ U$, so that $u<u^b$. By the continuity of $e(·)$ from, we may therefore choose $ε > 0$ small enough so that $u + ε<u^b$, and $e(p, u +
ε)<y$.

I don't understand the point below:

  Note that by definition of $v(·)$, $u ∈ U$, so that $u<u^b$.

$v(·)$ denotes the maximum utility that can be achieved for given prices and wealth, and I do not see why we cannot have $u = u^b$ such that $u ≤ u^b$ since $u^b$ may be finite. What am I missing here?

Even if $u^b$ is finite, it can never be achieved. This is what is meant by "does not attain a maximum". Rather, $u(x)$ approaches $u^b$ from below as $x \to \infty$. This is because $u$ is strictly increasing. If we had $u(x_*) = u^b$ for some $x_*$, then we would have $u(x_* + 1) > u^b$ and $u^b$ could not be a bound. This is why $U$ is written as the half-open interval $[u(0), u^b)$.


Offer curves in general equilibrium I'm having trouble understanding how to find the offer curves in general equilibrium. Is there a general way that we can use to find it?
I can understand the Pareto set and contract curve but when it comes to equilibrium I'm stuck.
For ex. if we have two consumers: Consumer 1 has as his initial bundle all of good $2:(0; 10)$ ;
while consumer 2 has all of good $1 : (10; 0)$.
Let their utilities be
$$u_1= min (x; y)$$
$$u_2= min (4x; 5y)$$
What would be the steps for finding the offer curves? Appreciate any help given.

In the picture below offer curve of individual 1 is given by lines connecting $E$ to $O_1$ and $O_1$ to $O_2$. And offer curve of individual 2 is given by lines connecting $E$ to $O_2$ and $O_2$ to $A$ and $A$ to $O_1$. Set of competitive equilibria are given by the intersection of two offer curves.
$p_X = 0$ and $p_Y = 1$ supports allocation at $O_2$ as competitive equilibrium. Allocations on the line segment $O_1A$ are supported by prices $p_X = 1$ and $p_Y = 0$.


Can a market function if everyone has the same preferences Assume everyone has the same preferences. Therefore there would be , for example, no trivial concept of liquidity preference differences etc.

Would the economy still work?

Even if everyone's preferences are exactly the same, as long as you leave diminishing marginal utility (e.g. each person would rather have nonzero proportions of apples and bananas rather than exclusively apples or exclusively bananas) and the advantage of division of labor (e.g. a farmer can produce more apples or more bananas if he focuses exclusively on growing one of those rather than trying to grow some of both), there will still be trade because people will want to maximize the value they accumulate by specializing their labor, but possess that value in the form of a wider variety of goods than their (efficiently allocated) labor produces.


How to analyze this guessing game? I sometimes play a guessing game with a friend. So fx we might try to guess how many people live in Russia. So person A says 100 milion and then person B says either 'more' or 'less'. If B says 'more' and in fact more than 100 milion live in Russia, then B wins and conversely. Is this game fair? I have an intuition that B probably has an advantage, but I'm not sure how to formally analyse the situation.

This is an interesting question!

It would seem that the game is not fair: Player A is at a pretty serious disadvantage.

Let's imagine each player has their own probability distribution of beliefs that is a sampling distribution of some population distribution centered at the true value. Both players are aware of this, and know their own distribution but have no further information on the other's belief distribution.

The optimal strategies are for player A to guess at the center of their own distribution and for B to guess "More" if the center of B's distribution is above A's guess and "Less" if it's below.

Player A will win if their guess is between the true value and Player B's guess.

-Since there is a 50/50 shot for either guess to be above or below the true value, 50% of the time the guesses will lie on either side of the true value and player B will win, regardless of whose guess is closer.

Via symmetry:

-25% of the time their guesses will be on the same side of the true value and player B's will be closer. Player B will also win here (deservedly).

-25% of the time their guesses will be on the same side of the true value and player A's will be closer. Here player A will win (deservedly).

Hence, player A will win only 1/4 of the time.

However the game also becomes increasingly complicated once you allow each player to have first order beliefs about the other player's beliefs (ie. specific information about the shape and center of the other's belief distribution), because then there's the opportunity for say, A to exploit their belief that B's guess is below the true value by guessing slightly lower than they would otherwise. And with second-order beliefs things get even more complex, and so on and so on. I'm not sure whether player A's disadvantage is robust to these nuances, but it's still quite striking to me that a game that at first glance seems fair is actually under normal conditions pretty lopsided!


Job Search and the Investment Problem I don't know what to take as the cost of accepting the job. Kindly guide me through deciding how to frame the equation.

You interview one employer per week, and your discount rate is $\delta = 1/(1+.07)$.  Suppose $x$ is the lowest quality job you would accept: anything worse than $x$, and you reject and keep searching. Let $J$ be the value of searching. Then we have the functional equation:
$$
J = \max_{x} \int_{x}^{1}z dF(z) + \int_{0}^x \delta J dF(z)
$$
The first term is the jobs you accept, since they are above your cutoff, and the second are the scenarios where you keep searching and forego the bad offer, so that you end up where you started tomorrow with an expected payoff of $J$. You have a uniform distribution so $dF(z) = dz$ on $[0,1]$.

Maximizing over $x$ gives the optimization condition
$$
-x*f(x*) + \delta f(x*) \delta J = 0
$$
so that $x^* = \delta J$.  Only if the offer is better than $\delta J$, the discounted value of continuing, do you accept; indeed, $\delta J$ is the opportunity cost of accepting, so I assume that is your $c$.

Now, let's solve for $J$. We have the equation
$$
J = \int_{\delta J}^1 zdz + \int_0^{\delta J}dz \delta J
$$
$$
J = \dfrac{1}{2}(1-(\delta J)^2) + (\delta J)^2
$$
$$
J = \dfrac{1+(\delta J)^2}{2}
$$
$$
0 = 1+\delta^2 J^2 - 2J
$$
This is a quadratic equation. The solution is
$$
J^* = \dfrac{1}{\delta^2} - \dfrac{\sqrt{4 - 4\delta^2}}{2\delta^2}
$$
$$
J^* = \dfrac{1-\sqrt{1 - \delta^2}}{\delta^2}
$$
and your search strategy is to accept if the job is better than
$$
x^* = \dfrac{1-\sqrt{1 - \delta^2}}{\delta}
$$
and reject otherwise.


Democratic elections of board members by employees? Would switching the selection of board members away from shareholders to employees (subject to bribery controls within campaigns, for salaries and stuff) limit the short-termism that the current corporate world seems to face?

In Sweden, 2 employees are members of the board in companies over 25 employees (by law). So at least here, not all board members are choosen by the shareholders.

One of the reasons many shareholders have stocks, is to influence the long term way the company is run. The main way to do this is by influencing which people are on the board. If big shareholders could not influence this, the cost of new capital into the companies would increase very much. Because with less influence there is a larger risk, and the way you pay for larger risk is by larger return on investments.

So in short, while it seems on the surface that it could limit the "short-termism", as you call it, it may actually have the opposite effect.