Dornbusch model for exchange rate undershooting Is it possible to reproduce the nominal exchange rate undershooting within the framework of the Dornbusch model?
If the government uses fiscal expansion, does undershooting happen?

This will be long, but the subject is worth it.

The gist of the "exchange rate overshooting" model in Dornbusch, R. (1976). Expectations and exchange rate dynamics. The Journal of Political Economy, 1161-1176. can be given using two equations:

Uncovered Interest Rate Parity (UIRP)
$$i  = i^*+ \dot e\tag{1}$$

where $i$ is local interest rate, $i^*$ is international interest rate, and $e$ is the logarithm of the foreign exchange rate, i.e. local currency units per unit of international currency basket. So an increase in $e$ reflects depreciation of the local currency. A decrease in $e$ reflects appreciation of the local currency (I would prefer to use the local exchange rate, where conceptually we link "rise" with "rise", but it is unfriendly to the phase diagram that it will follow). Then $\dot e$ is the expected percentage change on the exchange rate.

Money-market Equilibrium (in logarithms)

$$m-p = -\lambda i +\phi y \tag{2}$$

where $m$ is money supply, $p$ is the local price level, and $y$ is local output.

The assumptions are: international interest rate is fixed. There is perfect capital mobility, so the local interest rate is a "jump" variable, being able to adjust instantaneously (and so eq. $(2)$ holds always). Local output is at full-employment level. Local prices are "sticky": this means that the local price-level is not a "jump" variable anymore, but its evolution is governed by a difference/differential equation that reflects gradual adjustment. Money supply is long-term neutral.

The famous "overshooting" effect from an unanticipated permanent expansion of $m$ can then be derived: an increase in $m$ raises the left-hand side of $(2)$. Since prices are sticky (don't jump) and output is fixed, the only way that equilibrium can continue to hold in the money market is by a decrease in $i$ (due to the minus sign), so that that eq. $(2)$ continues to hold.
Given the fall in $i$, and the assumption of constant $i^*$, the only way now that $(1)$ can hold with a lower $i$ is for $\dot e$ to be negative, i.e. if the local currency is expected to appreciate. But raising the long-term neutral local money supply can only create expectations for local currency depreciation, not appreciation.
"Overshooting" cuts the Gordian Knot: the immediate response of the exchange rate $e$ is to depreciate by more (overshooting) than the eventual depreciation that is consistent with the increase in the money supply, and then it starts to appreciate to reach its new level (and so we can have $\dot e<0$).

The question asks about "fiscal expansion". How is this to be represented in the model? The only possible way is to assume that output can rise due to the fiscal expansion.

A fiscal expansion increases $y$ and raises the right-hand-side of $(2)$. Assuming that money supply is unchanged (and prices of course sticky), to equilibrate the money market, the local interest rate must now rise (to reduce demand for money balances that tend to rise to facilitate transactions on the higher $y$). Turning to $(1)$, with a higher local interest rate, we need expectations for currency depreciation ($\dot e >0$) in order for UIRP to hold. But a fiscal expansion, be it temporary or permanent, given a full-employment output or not, is usually associated with local currency appreciation (higher $e$).
So it appears that here too the overshooting effect obtains: the local currency jumps and appreciates by more than it is consistent with the fiscal expansion, and then it starts to depreciate, and we obtain the needed $\dot e >0$.

Let's put this in a (semi) phase diagram. We need a bit more specification, how the exchange rate and the price level change. Denoting $\bar e$ and $\bar p$ their long-run equilibrium values, we specify

$$\dot e = \theta(\bar e -e)\; \;\;\;\text{eq.}(3a)\\ \dot p = -v(p-\bar p)\; \;\;\;\text{eq.}(3b)$$

Notes: Both parameters are positive. The above look like ad-hoc "adaptive" schemes, but: the $\theta$ parameter can be determined so as to be compatible with rational expectations, the $v$ parameter is essentially determined endogenously, while the sluggishness of the price level is an  a priori assumption. This means that while both are governed optimally by a differential equation, the exchange rate can jump, while the price level cannot -it is "slaved" to its differential equation, while the exchange rate is only optimally characterized by its own equation, not chained to it.

By setting $\dot e=0$ we can obtain the expression for the long-run price level:

$$\bar p = m+\lambda i^* - \phi y \Rightarrow m = \bar p-\lambda i^* + \phi y\tag{4}$$
Inserting equation $(3a)$ into $(1)$, then $(1)$ and $(4)$ into $(2)$, we have

$$\bar p-\lambda i^* + \phi y -p = -\lambda[i^*+ \theta(\bar e -e)] +\phi y$$

Simplifying and re-arranging we get

$$ e = \bar e -\frac {1}{\lambda \theta}(p-\bar p) \equiv QQ\tag{5}$$

Equation $(5)$ is the saddle-path of the economy in the $e-p$ space (although in Dornbusch's paper time it was rarely called that). And it alone can tell us about overshooting. Note that the current exchange rate level depends positively on both its long-run value but also on the long-run price level. Following Dornbusch, we normalize the initial relative price of local to foreign output to unity, and so we have that $\bar e = \bar p$. This means that in the $e-p$ space the long-run equilibrium will be on the $45^{\text{o}}$ line. Putting $p$ on the vertical axis, the $QQ$ schedule will be downward sloping. Then the phase-diagram (without fixed-point loci, we don't need them since we have the saddle-path) is, for the case of a fiscal expansion,

We assume that we start at the long-run equilibrium point $A$, (and we will discuss this point later). Then, fiscal expansion happens. If it is permanent, it is associated with long-run currency appreciation and a lower long-run price level. This means that the saddle-path of the economy moves permanently to the left and becomes the $Q'Q'$ schedule. How can the economy find itself on the new saddle-path? It must jump. But it cannot jump vertically because prices are sticky. It can only jump horizontally: so it goes from point $A$ to point $B$ which is one the new saddle-path, and then starts to travel towards the new equilibrium, that is point $C$. But at $B$ corresponds exchange rate $e_B<\bar e_2$, the latter being the new long-run equilibrium exchange rate. And lower $e$ means more appreciated currency. So the exchange rate "overshoots" the appreciation level, and then starts to depreciate as the economy moves towards $C$.

Assume that the fiscal expansion is considered "temporary" (for decades, this was a good laugh in real-world macroeconomics -not so in the last couple of years). Still, even though the long-run equilibrium point remains $A$, there is short-term and mid-term horizons that must adjust. Here the point $C$ is some sort of "temporary" equilibrium point, which subsequently will start again to approach $A$. But the initial adjustment will happen exactly in the same way, and we will have overshooting.

Imagining a monetary expansion, amounts to shift the saddle-path outwards -and obtain overshooting related to depreciation.

A final note: if one plays around with the diagram, he will realize that
if we do not start at the long run-equilibrium level, and if we are sufficiently far away from equilibrium on the $QQ$ schedule, we may obtain undershooting in either case of policy. For the case at hand (fiscal expansion), imagine that we are at point $D$ on $QQ$ that at the time of the expansion corresponds to a price level below the one that corresponds to new equilibrium point $C$, and we are moving upwards towards $A$. When the fiscal expansion happens and the new saddle-path is $Q'Q'$ we will jump horizontally alright -but we will jump on the upward moving part of the new schedule (draw an horizontal line from $D$ to see that), i.e. we will undershoot the needed appreciation, and then we will gradually continue to appreciate the currency.
But this case is not unique to the fiscal expansion scenario -it can also happen in the monetary expansion scenario.

But if in principle the model permits everything to happen, why everybody went bonkers over Dornbusch paper back then, and it continues to be considered one of the high peaks of economic theory? For the historical framework one can read this. The fact that the model offered a clear and optimizing reasoning to account for the continuous ups-and-downs of the exchange rates (a new "bizarre" phenomenon at the time) I guess was reason enough (and also perhaps because open economies sufficiently far away from the equilibrium point of their nominal variables are not considered very likely -prices do adjust, after all).


What are the causes of increase in productivity of labour? This question has a specific context.
While reading Adam Smith's The Wealth of Nations, edited, with notes and marginal summary by Edwin Cannan in 1904, one comes across the following lines on the second page:

  The causes of this improvement, in productive powers of labour, and the order, according to which is produce is naturally distributed among the different ranks and conditions of men in the society, make the subject of the First Book of Inquiry.

Against "the causes" written at the outset in first line, there is a note, I presume by Edwin Cannan, which reads:

  Only one cause, the division of labour, is actually treated.

Here my aim is to know what other causes of increase in productivity of labour are known today (or were known to Edwin Cannan, back in 1904) that Adam Smith missed in his classic text.

In one word: Technology.

It's a bit abstract of course, but it summarises a lot of things very well. It's usually the A parameter in the models we have in growth theory (we augment e.g. Labour by Technology).
In empirical studies it has been tried to asses productivity by coming up with a measure called TFP (Total Factor Productivity).

A lot of specific things could potentially be subsumed under this broad term. Most importantantly it is the stock of knowledge (e.g. books, blueprints,...) that we improve and build upon. The growth of technology usually depends on the actual stock of knowledge in a certain economy. It can be achieved by Research & Development (in case the economy is on the technology frontier) or simply by technology adoption (e.g. in the case of catch-up economies).

Example: Knowing how to build a tractor, we only need one unit of labour (the guy who drives the tractor) to harvest, say, 100 kilos of grain per day instead of 50 workers that have to cut the same amount by hand. This would be a tremendous increase in productivity because the other 49 people can now do another job in which they potentially add to the economy's output. For building a better tractor (that drives with less fuel, can harvest more, ...) the economy needs the blueprint of the first tractor to build upon, and so on and so forth...


What is the difference between microeconomics money and macroeconomics money? In microeconomics, \$USD are a store of value.  I can sell a candy bar for \$1, and I can hang onto that \$1 for a year and then buy the same candy bar for that same \$1.  If I print \$1 (and get away with it), I can wait a year and buy a candy bar with it.  My \$1 has some inherent value.

In macroeconomics, \$USD are simply a means to inject liquidity into the market.  If the world sets up a global savings fund, and puts \$1 into it, that money is moot.  There won't be more candy bars next year than there are this year.  The \$1 is just a means to make my exchange of candy bars convenient.  It has no inherent value.

This is something of an over-simplification, and I'm no economist, so probably not entirely correct.  However, I frequently encounter people who apply personal-finance microeconomics-style reasoning to discussions of goverment fiscal policy.  I have found it to be incredibly difficult to introduce the above distinction, and I have no citations or sources.  This is just how it makes sense to me, and my words are apparently insufficient.

So, does this distinction have a name?  Is this distinction studied in any text and/or academic circles?  What are some ways that the microeconomic dollar and the macroeconomic dollar are two different beasts?  What are some simple ways to introduce this distinction to laypeople?

There is no distinction between money in microeconomics or macroeconomics. In both fields money is  medium of exchange, unit of account and store of value.

The misconception you have probably arises from the role money plays in microeconomics and macroeconomics.

In microeconomics money is almost always neutral - that is it has no impact on the real variables like equilibrium output, labor supply, demand etc. Many micro models even assume money away and prices are simply given relative to some numeraire good. There are exceptions of course.

In macroeconomics money is also assumed to be neutral in the long run but not in the short run. Hence in macroeconomics money is analyzed explicitly by including price levels, inflation etc. For example, as you mentioned one of the special effects of money is that it can provide much needed liquidity that can boost aggregate demand.

If you are looking into name for the distinction between cases where money matter and when not, that is when it is not neutral and when it is (both in micro and macro) the name is classical dichotomy although the concept is again mainly used in macro since in micro money is almost always neutral


The role of reinforcement learning in Economics While working on different research projects I got fascinated by RL which got applied to many fields that are focused on agent based modeling. Though it is a field in Machine Learning what interests me is the overlap between that apporach with problems in economics. The latest awesome paper I know is the "AI Economist" by Socher et al.. Do you know more fields/papers where RL is applied to economic decision making? And do you think that this approach can lead to a paradigm shift in economics ?

I would say applications of RL in economics are still in their infancy, though probably have applications in many state-based models, agent-based models, or fields with repeated interactions like game theory.
There has been some work done on what you might call the "microfoundations" part - which is studying "regret". Susan Athey, for example, does a lot of work in this space, and an example paper is "Policy Learning with Observational Data" found here: https://arxiv.org/abs/1702.02896
There are also some papers that shoes RL results converge to what one might obtain via rational expectations / equilibrium. For example, "Learning leads to bounded rationality and the evolution of cognitive bias in public goods games" by Leimar, O. & McNamara, J. M.


Contest: earliest written instance of economic thought Following part 5 of @Lateral fractal answer to "Handy Hints for Committing to the Beta" on the meta

  "Have fun! No fun = No play. No play = No beta. I'm sure I'm not the only one to feel that a fresh private beta is like unwrapping an early Christmas present."

and in order to complement the question "What was economics like as a field before Adam Smith, the father of modern economics?", I propose the following contest:

Let us try to elect the "best earliest" written instance of economic thought.

Answers should contain a short description of the work at stake with proper reference and dates, and votes by users will decide of the winner.

Without a single hope of success, I suggest that people vote according to the following criterion

$$ Vote_i(answer~a) = \begin{cases} 1, \qquad \text{if } u_i(e_a,d_a) \geq T_i \\   0, \qquad \text{if } u_i(e_a,d_a) < T_i\end{cases} $$

where $r_a$ is your subjective assessment of the degree of "economicness" of the described work, $d_a$ is the date at which the work was written, and $u_i$ and $T_i$ are a utility function and threshold of your choice.

Good luck to everyone!

I was recently amazed to discover instances of computational social choice in The Nine Chapters on the Mathematical Art, the Chinese counterpart of Euclid's element, written by several generations of scholars from the 10th up to the 2nd century BCE.

The core issues in social choice theory is the question of fair allocation, or fair collective decision. Once an allocation rule is devised, it remains to make it applicable. One of the topics of computational social choice is precisely to build efficient algorithm to compute the solution of allocations rules given some inputs.

This is precisely the topic of chapter 3 and chapter 6 of The Nine Chapters on the Mathematical Art, which are entitled (resp.)  "Proportional Distribution" and "Fair Levies". The two fair division rules considered are the proportional division and the weighted proportional division rule. These rules are taken as given. They are not justified in any axiomatic way in the book, which would be the standard practice in modern social choice theory. Yet algorithm are proposed to apply them which in my views makes it a legitimate early instance of computational social choice ( because it answer the question  "given this allocation rule, how can we compute its solution for any possible inputs?" at a time when the answer to this question was far from obvious for the allocation rules considered).

Some example problems you will find in chapter 3 :

(All the examples below are taken form the excellent commented edition of the Nine chapters by Kangshen Shen, John N. Crossley, Hui Liu Oxford University Press, 1999. This edition contains great references to instances of these problem in written sources from other civilization, such as in Euclid's Elements)

[Problem 1]

  " Now Given five officials of different ranks : Dafu, Bugeng, Zanniao,
  Shangzao, and Gongshi jointly hunting 5 deers. Tell : how many does
  each get if [the deer are] distributed according to their ranks?"

[Problem 2]

  "Now given a cow, a horse and a sheep have eaten up the seedlings of
  someone's field. The landlord demands 5 duo of millet as compensation.
  The shepherd says : "My sheep eats hals as much as the horse." The
  horse owner says : "My horse eats half as much as a cow." The
  compensation is to be distributed according to the rates. Tell : how
  much should each repay?"

An example problems you will find in chapter 9 :

[Problem 1]

  "Now given the task of transporting tax millet is distributed among four counties. County A, 8 days from the tax bureau, has 10 000 households; County B, 10 days from the bureau, has 9500 households; County C, 13 days form the bureau, has 12350 households; County D, 20 days from the bureau, has 12 200 households. The total tax millet is 250000 hu needing 10000 carts. Assume the task is to be distributed in accordance with the distance from the bureau and the number of household. Tell : how much millet should each county transport? How many cart does each county employ?"


The right term for a market that tends towards consumers switching suppliers freely Say there's a bunch of competing businesses, car repair shops for example, in a certain market. Business is good and there's a substantial average turnaround time for all of them. Someone decides they're going to add a bunch of capacity - hire more mechanics and expand their space.

Say that shop finds as soon as they add the capacity, customers figure it out, get in the shortest line available to them, and very quickly the capacity is all in use, and the turnaround time is only lower, slightly, as a result of the total supply in the entire market being greater. The shop's business volume has quickly increased in close proportion to the expansion, but that's about all that changed. What is the best term for what this observation says about the market? Low perceived switching costs? High market fluidity?

Say the opposite turns out to be true. The shop adds the same extra capacity, but the total number of people walking in the door doesn't change, or is sluggish to change. Turnaround time goes down, business volume stays the same, profits go down due to the unused capacity, and the competitors are humming along as though nothing happened. The managers are left thinking, "Either we have a reputation or marketing problem, or the market we're in is just ____." What's the term for a market that behaves this way?

If you want a general term, I would suggest frictionless market.

Normally, we would expect customers to reallocate themselves until the waiting time across shops is equalised. Indeed, all else equal, if the waiting time at $A$ is greater than that at $B$ then customers will switch from $A$ to $B$ until the discrepancy is eliminated.

If we do not observe such switching occurring then we say there is some sort of "market friction". Some examples of frictions:

switching costs: switching to a new supplier is costly and the gain in reduced waiting time is not sufficient to compensate consumers for the cost of switching;
informational friction: some consumers don't know of the existence of firm $B$ (or don't know how long its waiting time is). Perhaps they could acquire the missing information, but doing so takes time and effort, which is costly.
principal-agent problems: customers might rationally trust their current supplier because of positive past experiences (formally, reputation can solve moral hazard problems). Or perhaps they just had enough time to learn whether their current supplier is competent or not; something they don't know about the alternatives.
product differentiation: perhaps people like some products more than others. $B$ might be further from their house than $A$ so that visiting $B$ takes longer. Or maybe they know the mechanic at $A$ and like to talk to him/her. This makes them reluctant to switch to $B$.
behavioural factors: if consumers do/can not act rationally then they may fail to take advantage of potential gains from switching.

Here's a neat quote on the issue:

  As recognised by the 2010 Nobel prize, the study of market frictions has generated rich insights in many areas of economics, especially macroeconomics, labour economics and monetary theory.1 In industrial organisation, analysis has focussed on understanding how market frictions can create a source of market power by restricting consumers' ability to change suppliers (Wilson, 2012).


How is Singapore able to provide first-world public services, given its low income tax rate? Singapore has a relatively low personal income tax rate. Somebody who earns $80,000 after deductions only pay 3,350 in taxes, or about 4% effective rate (see 2012-2016 tax table). 80,000 is already well above the gross median income in Singapore, which is 3,770 per month in 2014 (source).

Despite, this Singapore is well known for its first-world public services, including public transportation, education, security, public housing and utilities. Personally I've lived there and I enjoyed the level of public services that otherwise can only be observed in countries with much higher taxation than 4% (e.g. United States, Canada or Western European countries) or countries that rely on oil revenues thus eliminating the need for taxation (e.g. Gulf countries). I don't know how to prove this with hard data, but taking a cursory look at International rankings of Singapore can probably convince you that this is the case.

How is this possible? Absent 'free revenue' such as oil, isn't high level of taxation necessary to fund such high level of public service?

I'm not an expert on this, but as a Singaporean, here are some factors off the top of my head (plus some Googling), explaining why Singapore is different from other first-world countries (in terms of revenue and expenditures).

In 2014, the personal income tax contributed only about 14.6% of government revenues (Source). In contrast, in Canada for example, it was nearly 48% of government revenues (C\$135.7b out of C\$282.3b in 2015).

One unusual revenue source deserves commentary: car taxes, including the so-called Certificate of Entitlement (COE). The COE is simply a piece of paper that entitles you to own a car. This recent news story mentions how the COE price hit a 5-year low, at "only" S\$45,002 (≈ US\$31,600) for small cars.

These COEs ("Vehicle Quota Premiums") make up 6.0% of government revenue.

Besides this very expensive piece of paper, a car-owner still has to pay all the other usual taxes (GST, road taxes) and insurance, plus a lot of unusual fees (e.g. Electronic Road Pricing - an idea which London borrowed). A lot of these flow to government coffers. Together, these make Singapore easily the most expensive place in the world to own cars.

"Public" housing is not cheap, heavily-subsidized housing.

It is true that over 80% of Singapore residents live in government housing (HDB flats). But these are not the sort of heavily-subsidized public housing you have in some Western countries. Indeed, it is a frequent gripe amongst Singaporeans that these are incredibly expensive and there are often conspiracy theories that the government sometimes actively works to push up prices.

Given the lack of transparency and also the fact that Singaporeans are actively encouraged to use their CPF (retirement) savings to pay for these homes, these conspiracy theories are not without merit.

Government housing is generally priced at market rates. For example, recently a 1,001 square foot flat went for S\$900,000 (≈ US\$632,000) - this was in the news because it broke a local (Clementi) record, but it doesn't get much cheaper than that.

There are some subsidies for first-time Singaporean home-buyers: If both you and your spouse are first-time buyers, then you get a total subsidy of S\$30,000. But this is not a large subsidy.

Some exceptional hardship cases (which involves an elaborate application process and onerous requirements) get to rent a small one-room flat for extremely cheap (as little as S\$26 a month), but these cases are so few in number as to be negligible.

Public transport is not a perennial loss-maker, but rather a profit-maker!

Unlike in say some US cities, where public transport is always losing money, public transport in Singapore ALWAYS makes very healthy profits. So these are not drains on but rather contributors to the government coffers.

If you've only been in US cities where there's often only a few people on a bus, you may find it hard to imagine. But come to any Asian city where public transport is always jam-packed and you can see why public transport is so profitable.

The CPF

This is a forced retirement savings scheme. In Sep 2015, the CPF had S\$293.9b in total members' balances (source). That's about 75% of GDP.

This is another frequent gripe of Singaporeans because these savings are all locked up (with very draconian rules on how you can gradually withdraw small amounts after you're 55).

Moreover, these funds are "managed" by the various Singapore sovereign wealth funds (GIC, Temasek). GIC in particular is not well-ranked in terms of transparency (source). So it is unclear as to how and how much exactly these help to contribute to Singapore government coffers.

Health care is very efficient (at least if compared to say the USA)

Healthcare spending is 4.6% of GDP (World Bank).

I have heard some of my friends who work at the Ministry of Health (Singapore) complaining about how inefficient the Singapore health care system is.

But as someone who has lived in both Singapore and the US for a good number of years, I would say (based purely on my own anecdotal experience) that the Singapore healthcare system is extremely efficient, especially if compared to the US. But the US is perhaps the worst-case scenario. I'm not sure how Singapore would compare to other first-world countries.


Is there anything like ratio between number of manufacturing industries to number of retail shops? In my country there are very few manufacturing factories,but large numbers of small retail shops (like one shop per house or may be even more). What are the effects in the economy of a country for having large numbers of retail shops,but very few manufacturing factories ?

Economists are generally more interested in the aggregate output of different sectors of the economy rather than the number of firms or outlets.  The difference between lots of small shops and a few large shops is quite different to the difference between the majority of GDP coming from industry or services.

If a country has a lot of small retail outlets and a few large manufacturing firms, that suggests that there are much larger economies of scale in manufacturing than retail.

The aggregate output of agriculture, industry and services are normally reported in GDP tables (here's an example for Rwanda).

A larger services sector is normally associated with richer, more advanced countries.  These images compare countries based on the relative size of industry, services and agriculture, and the proportion of the labour force working in each:

In poorer countries, agriculture tends to be lower productivity than other sectors, so those countries tend to show up as greener in the labour share than the GDP chart.


Why don't suppliers increase price after a positive demand shock? Forgive me, I am new to economics. Suppose there's a demand shock in the pork market and D1 shifts to D2. Why not keep quantity supplied fixed at 220 and then raise prices to $3.70, instead of moving from e1 to e2? It seems to me that suppliers stand to make higher revenues by only raising the price, because 3.70 * 220 > 3.50 * 228.

This is because of competition. If you would keep the quantity supplied low someone else would produce as they can both earn profit and increase their market share by doing so.
This being said there are situations where producers actually keep the quantity limited so they can increase the price. We call this kind of situation collusion (or cartel).
However, collusion is forbidden by laws in virtually all countries around the world so it is not easy to achieve. Firms can still try to collude tacitly, that is they try to collude without making specific deal which would be unlawful. However, tacit collusion is very hard to sustain and becomes harder the more firms there are in the market. This is because tacit collusion requires a lot of unspoken coordination among the firms (and it has to be unspoken as in many countries even discussing collusion is illegal and heavily penalized. Furthermore, many laws incentivize firms to ‘snitch’ as usually firms that snitch are not punished yet their colluding competitors are). You can see some industrial organization textbook that showcases models in which that can happen - for example Industrial Organization: Markets and Strategies by Belleflamme and Peitz or the Theory of Industrial Organization by Tirole. This is also touched upon in most introductory undergraduate economics textbooks when they discuss duopolies, so you might just try to look for duopoly chapter in book/lecture notes you follow, but they often don’t do the subject enough justice.
If you are however, studying economics 101 competition will be for most part assumed to be perfect and under perfect competition it’s virtually impossible to sustain any collusion agreement that is not enforceable in law (and again all such agreements are illegal in almost all countries).


Limitation of Unemployment rate What is the rationale behind government not including discouraged workers who have given up looking for jobs out of frustration given the state of economy?

This under-reports the job market and at the outset looks ridiculous.

The U-3 unemployment rates is the total unemployed, as a percent of the civilian labor force (official unemployment rate). By definition it excludes the discouraged workers, which, as Varun and Kent understand, is not true of all unemployed statistics. I liked what Karabell (2014) says about this statistic:

  ...[T]here is nothing wrong per se with the [U-3] unemployment figure,
  provided you know that the definition of employment is a construct,
  not a counting exercise....to be unemployed, according to the
  Bureau of Labor Statistics, you have to be in the workforce. To be in
  the workforce, you have to have a job or be looking for a job. To be
  looking for a job, you have to answer when surveyed that you have
  actively sought employment at some point in the past four weeks. If
  you haven't, you aren't actually unemployed. You are a discouraged
  worker, are marginally attached to the workforce, and are not counted
  as part of said workforce. And if you haven't been looking for a year,
  you're not, statistically speaking, even discouraged. You simply,
  statistically speaking, don't exist. You do, of course, actually exist
  (though that is a whole other topic), but not in the statistical
  universe of employment data.

  The result of such methods is that it is possible, and not just
  theoretically, for the unemployment rate to drop not because more jobs
  were created or more people were hired but simply because more people
  gave up looking for work and dropped out of the workforce. That is, in
  fact, What happened in the United States and swaths of Europe after
  2009, though in America, there were also real job gains in 2010 and
  after. Nonetheless, the unemployment rate, which receives so much
  public scrutiny, went down in part not because the economic system was
  creating so many new jobs but because it was failing to do just that.
  To be fair, the BLS keeps multiple unemployment statistics and
  different unemployment rates, including ones that include
  underemployed, marginally attached, and part-time workers, but there
  is only one headline number. That is what the BLS points to in its
  monthly release; that is what the media pick up; and the other numbers
  are at best footnotes.

The Leading Indicators: A SHort History of the Numbers That Rule Our World
Zachary Karabell (2014)

Therefore, if your question is why does U-3 exclude discouraged workers, the answer is that's an essential part of the definition of discouraged.

If your question is why did they define U-3 this way in the first place, I am not sure. I suspect the following. The definition of the labor force, which is workers and those actively looking for work or waiting to be recalled to work. Detached and marginally attached workers are not in the labor force. Once you defined the labor force, a natural question is what fraction of the labor force is not working, and that statistic is U-3.

If your question is why is this statistic more widely discussed than other economic statistics, the answer may simply be that it is older. The U-3 statistic goes back to 1948 while U-4, U-5, and U-6 start in 1994. People may be used to talking about the better established statistic and enjoy the longer time series, and see insufficient benefit to switching to the newer statistics, despite the benefits. A sort of Dvorak vs. Qwerty keyboard issue for economic statistics.

In a way, this is like an issue of GDP measurement. Home production is not in GDP and Robert Barro has argued that investment is double counted in GDP. If you want to use GDP as a measure of something else, these may be big problems and other measures may serve better. However, if you want to compare GDP across countries or time, these issues matter less than having something useful and consistent to compare them with. Commenter ABC at Marginal revolution has a nice quote about this:

  The algebra is fine as it is, but I take strong issue with the idea
  that GDP "mismeasures" something. As any decent intro macro course
  will tells its students, "GDP is not a measure of welfare." Indeed,
  the Kuznets quotes that Barro points to are exactly of this nature; we
  have this high profile GDP number, but it doesn't do what you want it
  to do - or at least not ALL the things you want it to.

  What Barro does here is construct a new measure - present discounted
  value of consumption - and shown how it relates to GDP (and GDP's
  present discounted value). In addition, he provides a capital income /
  labor income decomposition for both measures (GDP having such an exact
  decomposition because of constant returns to scale, Euler's identity
  for homogenous functions, and marginal-value input prices). He then
  says present discounted value of consumption, and its associated
  decompositions, are "right" while the other the construction GDP is
  "wrong" and thus gives "misstated" decompositions. Of course,
  "right/wrong" and "correct/incorrect" begs the question - right about
  WHAT? Correct for WHAT?

  Barro says that in present discounted value terms GDP "double counts"
  investment. The "double counting" is only relative to how things are
  calculated for the present discounted value of consumption. This just
  reflects the fact that GDP counts production, while consumption only
  counts consumption.

ABC's comment on Marginal Revolution Post: Robert Barro says we double-count investment


Monopolistic Competition- productive efficiency In my textbook, it says that monopolistic competition will never lead to productive efficiency (where $MC=AC$) in the short run and long run. I understand that they are meant to be producing at the profit-maximising level of output (where $MC=MR$), but all the diagrams showcase scenarios where the $AC$ curve is always positioned such that the points where MR=MC does not intersect where $MC=AC$.

My question is, is there any possibility that the profit-maximising level of output will also happen to be where we've achieved the lowest possible unit cost (lowest $AC$)? So that $MC=MR=AC$, and the firm can be both productively-efficient whilst also being profit-maximising? Why is this not possible?

Let $q$ denote the output of the firm, and let $\varepsilon_p(q)$ denote the elasticity of price w.r.t. quantity sold. We know that when profit is maximized
$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)}.
$$
We also know that in the long-run equilibrium firms have zero economic profit, i.e.
$$
AC(q) = p(q).
$$
Productive efficiency is achieved when $AC(q)$ is minimized, and at that point
$$
AC(q) = MC(q)
$$
holds as well. Combining these we get
$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)} = \frac{AC(q)-AC(q)}{p(q)} = 0.
$$
So the only case when all these properties coincide is when the quantity sold has no effect on the market price, i.e. when demand is perfectly elastic.

An explanation without the elasticity part:
From the other two properties (long-run equilibrium and productive efficiency) we have
$$
MC(q) = AC(q) = p(q).
$$
Together with profit maximization this implies that firms are price takers, $MR(q) = p(q)$, but since
$$
MR(q) = \frac{\text{d} \ p(q)q}{\text{d} \ q} = p(q) + p'(q)q
$$
this is only possible if $p'(q) = 0$, which again means that demand is perfectly elastic.

All the above assumed that the demand and inverse demand functions are continuously differentiable. You could have a situation with not perfectly elastic but kinked demand, where profit maximization occurs exactly at the kink. In this case $MR(q)$ does not exist at the point of profit maximization.


Debt bearing capacity In times of low yields and the prospect of rising yields, I asked myself if the european countries can ceteris-paribus cope with rising yields. Is there a way to calculate the effect of rising yields (ignore growth etc. for the time beeing), calculate a break-even yield where the debt burden becomes to much to cope. I thought about calculating the effect via government bond indices and the primary budget. Are there research papers about this?

There’s a huge literature on this topic. I would start by looking at the IMF Fiscal Monitor, and see what is interesting. Their research papers will refer to other papers that can be pursued. IMF Fiscal Monitor page.

Otherwise, you can search for papers on “fiscal sustainability.” As a disclaimer, I am not a fan of most the literature. This paper by Scott Fulwiler discusses interest rates and sustainability, and it critiques some of the technical issues in the literature - link to paper.

In any event, there is no “magic tipping point” for default. It is up to the government involved to decide at what point interest costs are excessive, and whether they will default rather than keep going. There have been almost no default of floating currency sovereigns, so there is no data set to work with. (Most sovereign defauits have occurred when they borrow in another currency; the euro area is discussed below.)

The euro area countries face a particular problem that is not really an issue for other developed countries: they do not have control of their domestic interest rates. Their debt may be issued at high spreads versus other euro interest rates, and so a default becomes a self-fulfilling prophecy (as seen in Greece). Most of the “fiscal sustainability” literature assumed that a country controls the interest rate it borrows at, so the situation is much less risky.


Can PPP adjusted values be compared over time? I'm a little baffled about this:
Let's say I want to compare the average income of 2 countries, A and B. Of course I'm interested in the real income so I adjust to PPP using the big mac index of that given year (dividing the average salary by some constant).
Now I do it again, but 10 years later (with the big mac index of t = 10).

Question -
Can I compare the adjusted average salary of country A, t=0  to the adjusted average salary of country A, t=10?

One can argue that no, because the index used for PPP adjustment changed over time, so we need to take account for this. That's why when you adjust for inflation you have specify to which year the prices have been fixed.
You can't compare a real salary adjusted to 2003 prices to a real salary adjusted to 2013 prices.

My logic says yes, because assuming that McDonalds hasn't changed their Big Mac over the past 10 years, the unit of choice remained that same - one delicious burger and it's NOT like adjusting to inflation (in which you have to randomly assign a base year).

There are two questions here:

Whether it is meaningful to compare PPP-adjusted values intertemporaly

Whether the "Big Mac" index is a good index for PPP-adjustment.

I will occupy myself with the first question.
Why are we doing PPP-adjustment? We do it to make inter-country comparisons against a benchmark country (usually the US). So this country's magnitude remains unadjusted. Note that what we adjust is nominal magnitudes not "real" ones (that are magnitudes adjusted of in-country inflation).
So at time $t$, and using country $A$ as the benchmark, we can obtain the PPP-wage of country $B$, say $\bar w_{B|A}(t)$. We could think of that as "I live in country B. My living standard at time $t$ is as if I was living in the benchmark country $A$ and my wage was $\bar w_{B|A}(t)$". And comparing that to the actual (nominal) average wage in country $A$, $w_A(t)$, we get a measure of the living standards in country $B$ compared to those in the benchmark country $A$.
Consider now the PPP-adjusted wage from the same country at time $t+k$. It says " I am living in country B in period $t+k$. My living standard at time $k+t$ is as if I I was living in the benchmark country $A$ and my wage was $\bar w_{B|A}(t+k)$".
It is evident that in order to meaningfully compare $\bar w_{B|A}(t)$ and $\bar w_{B|A}(t+k)$, we must deflate the latter by the price deflator of the benchmark country A, covering the period $(t,\, t+k)$. Doing that, makes the two comparable.
But the main purpose of PPP-adjustment is inter-country coparisons, not same-country intertemporal comparisons. By considering the ratio in two different time periods,
$$\frac{\bar w_{B|A}(t)}{ w_A(t)},\qquad \frac{\bar w_{B|A}(t+k)}{w_A(t+k)}, $$
we can see whether living standards have converged or diverged.
A discussion on recent PPP issues can be downloaded here.
A visual tool can be accessed here.


Is it possible that the minimum point of a short run cost curve does not touch the long run cost curve? In the answer to this question, the answerer said "the minimum point of a short run cost curve will be above the long run cost curve". Is it true? If so, how would it be so?

I thought that if e.g. the short-run capacity is too small or too big, it would just be represented as different short-run cost curve tangent with the long-run cost curve.

The idea that the long run average cost curve (LRAC) must pass through the minimum points of the short run average cost curves (SRAC) is a fallacy, but it seems to be a remarkably plausible one.  It was the source of a famous error by the economist Jacob Viner, referred to in this paper by Silberberg.  Underlying the fallacy is perhaps an assumption that the points of tangency with the LRAC must be the minimum points of the SRAC’s.  These points are coincident in the special case of a SRAC tangent to the LRAC at the minimum point of the latter.  But usually they are distinct, as in the numerical example below.

Suppose a firm has a Cobb-Douglas production function with increasing returns  $y = x_1^{0.6}x_2^{0.6}$.  The inputs are bought in markets in which supply is not perfectly elastic, so that costs are increasing functions of quantities:

$\quad c_1(x_1)  =  20x_1 + x_1^2$

$\quad c_2(x_2)  =  20x_2 + x_2^2$

The symmetry between the two inputs in respect of both technology and costs is not necessary to obtain a suitable example, but it is convenient because it implies that every point on the LRAC must satisfy $x_1=x_2$ (see Appendix below for proof).  This simplifies the derivation of  the equation of the LRAC.

LRAC

Writing $c(a,b)$ for the total cost function with inputs $a, b$ and given $x_1=x_2$ we have:

$\quad c(x_1,x_2) = 40x_1 + 2x_1^2\qquad(1)$

$\quad y = x_1^{1.2}\qquad(2)$

Hence:

$\quad x_1 = y^{5/6}\qquad(3)$

$\quad c(x_1,x_2)  =  40y^{5/6} + y^{5/3}\qquad(4)$

and so:

$\quad LRAC = \frac{40y^{5/6} + 2y^{5/3}}{y}  =  40y^{-1/6}+2y^{2/3}\qquad(5)$

To find the minimum point:

$\quad \frac{dLRAC}{dy}  =  (-40/6)y^{-7/6} + (4/3)y^{-1/3} = 0\qquad(6)$

$\quad -40/6 + (4/3)y^{5/6} = 0\qquad(7)$

$\quad y^{5/6} = 5\qquad(8)$

$\quad y=6.90\qquad(9)$

To confirm this is a minimum:

$\quad \frac{d^2LRAC}{dy^2} = (280/36)y^{-13/6} + (-4/9)y^{-4/3}$
$\quad = (280/36)0.0152 – (4/9)0.0761 = 0.118 – 0.034 = 0.084 \boldsymbol{> 0}\qquad(10)$

The inputs at this minimum, using (3), are:

$\quad x_1 = x_2 = 6.90^{5/6} = 5.00\qquad(11)$

SRAC

Suppose now that $x_1$ is freely variable but $x_2$ is fixed in the short run at a value other than $5.00$, say $2$.  Then:

$\quad y = x_1^{0.6}(2^{0.6})\qquad(12)$

$\quad c(x_1,x_2)  =  20x_1 + x_1^2 + 44\qquad(13)$

Hence:

$\quad x_1 = (2^{-0.6}y)^{5/3} =  (1/2)y^{5/3}\qquad(14)$

$\quad c(x_1,x_2) = 10y^{5/3} + (1/4)y^{10/3} + 44\qquad(15)$

and so:

$\quad SRAC(x_2 = 2) = \frac{10y^{5/3} + (1/4)y^{10/3} + 44}{y} = 10y^{2/3} + (1/4)y^{7/3} + 44y^{-1}\qquad(16)$

The first derivative is:

$\quad \frac{dSRAC}{dy}  =  (20/3)y^{-1/3} + (7/12)y^{4/3} – 44y^{-2}\qquad(17)$

Relation between LRAC and SRAC

The two curves meet when $x_1=x_2=2$ implying $y = 2^{1.2} =  2.2974$ since at that point, using (5) and (16):

$\quad LRAC = 40(2.2974^{-1/6}) + 2(2.2974^{2/3}) = 34.822 + 3.482 = \boldsymbol{38.30}\qquad(18)$

$\quad SRAC = 10(2.2974^{2/3}) + (1/4)(2.2974^{7/3}) + 44(2.2974^{-1})$
$\quad = 17.411 + 1.741 + 19.152 = \boldsymbol{38.30}\qquad(19)$

Moreover they are tangential at that point since using (6) and (17) the respective slopes are:

$\quad \frac{dLRAC}{dy} = (-40/6)(2.2974^{-7/6}) + (4/3)(2.2974^{-1/3})$
$\quad = -2.526 + 1.010 = \boldsymbol{-1.52}\qquad(20)$

$\quad \frac{dSRAC}{dy}  =  (20/3)2.2974^{-1/3} + (7/12)2.2974^{4/3} + (-44)2.2974^{-2}$
$\quad = 5.052 + 1.768 – 8.336 = \boldsymbol{-1.52}\qquad(21)$

However, this point of tangency is not the minimum point of the SRAC.  Using (17) to find the minimum:

$\quad \frac{dSRAC}{dy}  =  (20/3)y^{-1/3} + (7/12)y^{4/3} – 44y^{-2}  =  0\qquad(22)$

$\quad (20/3)y^{5/3} + (7/12)y^{10/3} – 44 = 0\qquad(23)$

Treating this as a quadratic equation in $y^{5/3}$, or by trial and error, it can be found that $y$ is approximately $2.525$.  To confirm this is a minimum:

$\quad \frac{d^2SRAC}{dy^2}  =  (-20/9)2.525^{-4/3} + (28/36)2.525^{1/3} + (88)2.525^{-1}  =  -0.646 + 1.059 + 34.851 = 35.26 \boldsymbol{> 0}\qquad(24)$

At this minimum point:

$\quad SRAC = 10(2.525^{2/3}) + (1/4)2.525^{7/3} + 44(2.525^{-1})$
$\quad = 18.543 + 2.170 + 17.426 = \boldsymbol{38.14}\qquad(25)$

This is lower than the point of tangency with the LRAC ($\boldsymbol{38.30}$), but above the LRAC at $y = 2.525$ which using (5) is:

$\quad LRAC = 40(2.525^{-1/6}) + 2(2.525^{2/3}) = 34.278 + 3.709 = \boldsymbol{37.99}\qquad(26)$

Appendix

Suppose $x_1\neq x_2$ and let $x* = \sqrt{x_1x_2}$.  Then:

$\quad y(x_1,x_2) = (x_1x_2)^{0.6} = (x*^2)^{0.6} = y(x*,x*)\qquad(27)$

$\quad c(x_1,x_2) = 20(x_1 + x_2) + x_1^2 + x_2^2$
$\quad = 20[(\sqrt{x_1} - \sqrt{x_2})^2 + 2\sqrt{x_1x_2}] + (x_1 – x_2)^2 + 2x_1x_2$
$\quad \boldsymbol{>} 2[20\sqrt{x_1x_2}) + (\sqrt{x_1x_2})^2] = c(x*,x*)\qquad(28)$

Thus the input combination $(x*,x*)$ yields the same output at lower cost than $(x_1,x_2)$, and so the latter does not correspond to a point on the LRAC.


Does a 2sls model use logarithm? In an ols model demand ~ price , is it reasonable to use log-log for price and demand, but in a 2sls model does the logic remain the same?
I found articles simply using 2sls without using a log to find the equilibrium, does that make sense?

You are allowed to use logs in 2sls model. The reason why that article does not use logs is that the article assumes linear supply and demand.
At the beginning the author states:

Consider the following supply and demand functions
Supply:$Q=\beta_1 P + \epsilon_s$
Demand: $Q=\alpha_1 P+ \alpha_2 I+ \epsilon_d$

If the author would instead assumed supply and demand to have following form:
$Q=P^{\beta_1}$ and $Q=P^{\alpha_1}I^{\alpha_2}$ then taking logs would make sense but if the supply and demand are already assumed to be linear taking logs is unnecessary as you already have the right functional form and you can just directly apply linear model without log-linearizing it.
For example, Soderbery (2015) who estimates import supply and demand elasticities uses logarithms and models his supply-demand system using 2sls model with log-log form.


How does private benefit exceed social benefit? I've done economics for 3 months only, so bear with me.

If Social Benefit = Private Benefit + External Benefit

Then how is it physically possible that private benefit could ever exceed social benefit?

Someone told me that if there is an External Cost this reduces the social benefit, but why? If the Social Cost already takes the External Cost into account, why would the Social Benefit do the same?

If the Social Benefit and Social Cost are two different things then why do they both take into account costs and benefits?

I really do not understand the logic behind it and do not understand my teacher's explanation. Much help is needed.

Thank you.

The net social welfare can be written as $W=B-C$, where $W$ is total welfare, $B$ is the benefit, and $C$ is the cost.

Suppose we are interested in thinking about a negative externality that implies a social cost of $E$. Then we would have $$W=B-(C+E)=B-C-E.$$

Imagine if, instead of viewing this as a social cost, we interpret the externality as reducing the social benefit. The expression would then be $$W=(B-E)-C=B-C-E.$$

So the two ways of thinking about the world are completely equivalent and it is merely a naming convention whether we call the externality a cost or a reduction in benefit. I guess your confusion has come about because you heard people on different occasions using the different conventions, without being clear about which convention they were using.

The only thing that matters is that we don't double-count the externality by including it as both a cost and a reduction in benefit.


Everyone has the same marginal rate of substitution I'm currently reading Varian's Intermediate Microeconomics and what struck me, is this statement on page 89 of the 8th edition.

  If everyone faces the same prices for the two goods, then everyone will
  have the same marginal rate of substitution, and will thus be willing to
  trade off the two goods in the same way.

I mean, I see the mathematics behind it, but it still seems kind of hard for me to grasp intuitively. Could someone provide me with examples or another explanation to clear things up ?

There are a couple of assumptions driving this result. For instance, all agents are assumed to have preferences and a budget such that their optimal bundle is interior. For example, budget constraints were people cannot afford food are ruled out. Likewise, situations, where someone does not like one of these goods or has very strong substitutions patterns between a pair of goods, are also ruled out.  All preferences are assumed well-behaved (monotonic and strictly convex).

Secondly, the market is competitive, so all people face the same prices not only for tomatoes but also for airplanes, wages, etc. This almost never happens in reality, but in some cases, it is a good approximation. For example, when considering shopping for breakfast, the relative prices between cereal and yogurt and eggs roughly reflect the willingness for every person to substitute one good for another.

This is a striking result and I see it more of a theoretical benchmark whose main intuition is that prices contain a whole lot of information about how people are willing to substitute one good for another. When they don't, it is very interesting to understand why.


Income/Wealth Inequality; Why does it matter? I have attempted to answer this question on my own through reading books, blog, and other economic pieces of literature but they all fail to articulate why wealth or income disparities matter. Now obviously leveraging slave labor or paying someone literal pennies is an issue. But why does the disparity of wealth between say the middle class and the top 1% matter?

From what I have studied it seems cultivating a competitive market (fair prices for goods and housing), managing inflation, and ensuring a strong currency is really what is necessary. Given these things everyone would be able to buy and sell goods or services and still be able to squirrel some money away for retirement. What am I missing?

There could be some complicated models where high inequality could have effect on efficiency - the research is inconclusive (or to be more precise almost non existent) on the matter.

However, in Public Economics, depending on the selected welfare function (ralwsian, utilitarian, charitable conservative/libertarian etc.), inequality matters for optimal taxation. For example, even in competitive market without any imperfections someone’s marginal product of work and hence wage might be close to zero or not enough to afford food. From pure efficiency perspective that would be optimal, but under Ralwsian/utilitarian or also some other welfare function there should be some transfers to said person.

This being said economics does not provide any answers into which welfare function is the right one for that you need to study philosophy. Inequality as far as currently understood is a matter of taste.


Why not create money to assist the post-COVID-19 economy? In an imaginary country, a horrible computer virus infected some banks.
Within a week, it caused the wealth of a large portion of the population (say, 50%) to drop significantly.
For some, it cut 80% of their checking account and savings.
Others - it put into debt.
The losses were great and posed a threat to the country's economy.

The following month, security experts analyzed the effect of the virus, and identified the majority of affected accounts, and the magnitude by which they were affected.

Then, the government ordered the banks to revert these people's accounts to their state right before the virus has manifested itself.

Calamity was averted, and the economy restored.

As I understand, the action of reverting the accounts' state can be seen as "generating money". Pushing large amounts of "free" money into a market causes inflation. But I think that in the above scenario it shouldn't create inflation, because the generated money substitutes money that has vanished due to an external event.

My questions are:

From economic perspective, does the action of reverting the affected accounts' states make sense? Are there downsides?
If it makes sense, then how is it different from the widespread loss of income incurred by the COVID-19 pandemic? If a business was forced to shut down for two months, during which it had no income, but had to keep paying rent, paying salaries, etc - why shouldn't the government generate and provide a sum that would cover this business' expenses for that period? Not get the money as a loan, but just print it. Wouldn't this solve the problem for free, without negative consequences?

First, the two examples you list here are not equivalent. In the first example some portion of money circulating in economy is destroyed by the computer virus so reverting it would just keep amount of money in economy constant whereas in the second case you actually expand money supply because people 'loosing money' due to paying rent etc does not mean money in economy is being destroyed. For example, if an economy would have $\$100$ and the computer virus would eliminate $50\%$ leaving the economy with $\$50$, printing additional  $\$50$ would just bring the money supply in economy back to $\$100$. However, when people 'loose' money because they are paying rent, buying food and so on money is not being destroyed but just transferred from person A to person B. If I have $\$100$ and pay $\$50$ rent money supply is still $\$100$,  printing additional $\$50$ would increase it further to $\$150$.

The above being said it actually makes sense to expand money supply during recessions but it is not consequence free. Money in itself is not source of wealth - production of goods and services is where wealth comes from and money is just a measuring stick for its value. If you expand the amount of money in economy and the expansion is believed to be permanent then it will eventually lead higher prices since more money will be chasing the same amount of goods and services leading to inflation. Society cannot make itself richer just by printing money.

However, inflation is actually what you want during recession because there are many market rigidities such as sticky wages which can make recession worse than they are. In recession you want wages to fall to prevent unnecessary high unemployment, but people often have long term contracts with stipulated nominal wages that would take time to renegotiate. Inflation helps to 'grease the wheels of economy' because by reducing the value of money it reduces the value of peoples' wages in real terms.

Furthermore, in recession many business might face a liquidity problems - they might be overall healthy and profitable but might have hard time to have enough cash at hand. In that kind of situation you would again want to expand money supply in order to make more cash avaiable for them so they do not need to fail unnecessarily.

For these and other reasons central banks around the world are actually expanding money supply now, although they dont do it in a direct way as you described, but rather indirectly through slashing interest rates and buying bonds (see for example how Fed responds to covid19 here). The reason why they do it in this indirect way is that many experts think that if central banks would be just sending money directly to people it would politicize them, but central banks work best when they are independent technocratic institutions. Moreover, some central banks around the world are not even allowed to just do that due to the rules under which they are set up. However, central banks still can do that indirectly by buying government bonds and letting politicians decide on how to organize transfers.

The above being said while monetary expansion during recession helps the economy, the monetary expansion itself does not make economy any richer. This is especially so in situation where the problem is that economy is actually not allowed to produce goods and services in many sectors due to lockdown. Monetary expansion does not solve that underlying problem in any way. If government would just continue sending cash transfers to households payed by expansion of money supply eventually you would just get an inflation and prices would just rise higher and higher. This would have several downsides of eroding peoples savings, at higher levels inflation could also increase uncertainty, it would create menu costs as business would be forced to change their prices too often,  shoe leather costs as people would hold less then optimal cash balances, it would create fiscal drag by pushing people into higher tax brackets, it would redistribute income from lenders to borrowers and so on. So it cannot be done just without any cost.


Ratios to determine company's current purchasing power what are the parameters which will affect a company ability to buy a new product based on its current financial situation.  Is there any metrics or ratios which will tell the current company purchasing power?

You could probably get a very good idea of a company's ability to spend money - that is, make capital investments, develop new ventures etc. - by investing its balance sheet and develop both liquidity and leverage ratios. High liquidity with low leverage means that a company is probably not investing enough. Low liquidity with high leverage ratios would indicate a company has already invested heavily and is not in a position to undertake new ventures etc.

Here is a good introduction to leverage ratios
 Here is a good introduction to liquidity ratios


price space as a dual of commodity space I often see papers in economic theory mentioning a duality between the price space and the commodity space. I think they refer to duality of vector spaces. I.e, the commodity space is a vector space, a price system is a linear functional over the commodity space (assigning a value to each bundle), and so the space of all possible price systems  is the dual of the commodity space.

Is there a standard reference that explains this duality in more detail?

If you want to see a nice usage of duality in economics, in Mas-Colell, Whinston, and Green  there's a nice section on duality in the second or third chapter when talking about Hicksian demand. The treatment is pretty straightforward if the consumption space is real and has a finite dimension. I think understanding that basic usage of duality is illustrative of its general utility.

If the idea is to understand what duality is and how it operates, I think Conway is a great book.

(Just undergrad here, hope I'm not mixing up topics).


Is this supply and demand reasoning in Picketty's book circular? In his hugely successful book, Thomas Piketty writes:

  To be sure, there exists in principle a quite simple economic mechanism that should restore equilibrium to the process: the mechanism of supply and demand. If the supply of any good is insufficient, and its price is too high, then demand for that good should decrease, which should lead to a decline in its price.

This is circular, because a decline in price $p$ would again lead to an increase in quantity demanded $q$ which would again contribute to a situation of excess demand, right?

$\text{Excess  demand} \implies p\uparrow \implies q\downarrow \implies p\downarrow \implies d\uparrow \implies \text{Excess demand}$

Is Mr. Piketty forgetting to consider a possible supply-side reaction to the described excess-demand situation situation? What could be the reasons not to include this? Does it serve his argument? Is it economically as untenable as it appears?

The quote describes a very specific situation, which is not the general description through which the "law of supply and demand" is presented in non-technical terms.

It says

  If the supply of any good is insufficient

The correct term would be "quantity supplied at a specific point in time". Then the "insufficient" part, supposedly means "quantity supplied is less then quantity demanded". Ok. Then it says

  and its price is too high

So not only suppliers bring in the market a lower quantity that the quantity demanded by the consumers, they also "charge" a "too high" price... "too high" with respect to what? Presumably the consumers' "willingness to pay"?

So the first part of the quote describes consumers that have a priori specified both quantity desired and maximum acceptable unit price: "I want to buy 4 kgr of bread but at a price of no more than 0.5 euro/ Kgr." I understand that this sounds familiar: we tend to determine our needs/wants both in terms of quantity and we have some idea about what a "reasonable price" could be, by our past experience.

But once a consumer has specified two aspects distinctly, it must also decide on priority, if any: if the price is above 0.5 euros/Kgr, will he buy nothing? if this is case, then this consumer has "lexicographic preferences" over the unit price: he cares first for the unit price, and then for satisfying his needs through the consumption of bread. If the unit price is not right, he won't buy at all. This is a special situation, not just theoretically, but as to the degree it represents the "majority" of consumer behavior as observed in practice.

But ok, let's analyze the special situation presented in the quote.

The bakery has only 2 kgr and sells for 1 euro/Kgr. I won't buy at all, because the unit price is not right. I don't care whether the baker brings in 2 more Kgr to reach 4 - as long as "the price is not right", I am not buying. So for this special case, "supply responses" are irrelevant. Now if the baker lowers the price to 0.5 euro per Kgr, I will buy his 2 kgrs, (Piketty's quote ends here)
and I will tell him that I am willing to buy 2 more Kgrs as long as he will sell these 2 additional Kgrs also for 0.5 euros. If the baker agrees, "supply responds" and the price is not raised back. If he brings in the 2 additional Kgrs but raises the price above 0.5 euro/Kgr, they will be left unsold, and we will have excess quantity supplied, not excess quantity demanded.

What would be a more  general framework?

"I will spend 2 euros for bread. if the price is 0.5 euros/Kgr, I will buy 4 kgr. If it is 1 euro/Kgr, I will buy 2 Kgr. Etc". In other words, through its utility maximization, the consumer will determine the budget share that he will allocate to bread, and quantity demanded will be determined by the price.
Assume that there exists also a second consumer, that he has decided to allocate 4 euros for bread.
The baker appears with 4 Kgrs charging 1 euro each.
The first consumer asks 2 kgrs and the second asks for 4 kgrs, for a total of 6 Kgrs. We have excess quantity demanded, but not "too high a price".
Now, the baker is not in our heads, he does not know the budget shares we have allocated for bread: he thinks "hey, these guys ask for more than I have, so it may be the case that I can still sell my 4 Kgrs for a higher unit price, -they seem to really want that bread so maybe they are willing to pay a higher unit price".
He then declares that the price of bread is now 2 euros/Kgr. The first consumer then asks for 1 kgr, the second asks 2 kgr, for a total of 3 Kgrs. Now we have excess quantity supplied. This is how the "cobweb" theorem works bringing market equilibrium.

Of course the baker could conceivably think "hey, let's make 2 more Kgrs, since these guys ask for 6 Kgrs in total, keeping the price at 1 euro/Kgr". Whether he will choose to sell his 4 Kgrs for 1.5 euro/Kgr, or 6 Kgrs for 1 euro/Kgr, I will leave for the OP to contemplate.


How to write the dual problem of this maximization problem? On this note for  Shapley-Shubik model, there's a maximization problem:

$$\max_{x_{ij} \in \mathbb{R}^{M \times N}}\sum_{j \in N}\sum_{i \in M}v_{ij}x_{ij}$$
$$\text{s.t.}  \ \sum_{j \in N}x_{ij} \leq 1 \ \forall i \in M$$
$$\\ \sum_{i \in M}x_{ij} \leq 1 \ \forall j \in N$$
$$x_{ij} \geq 0 \ \forall i \in M\forall j \in N$$

The dual problem of it is

$$\min_{s_j \in \mathbb{R}^N, p_i \in \mathbb{R}^M }\sum_{j \in N}s_j+\sum_{i \in M}p_i$$
$$\text{s.t.} s_j + p_i \geq v_ij \ \forall i \in M\forall j \in N$$
$$\\ s_j, p_i \geq 0 \forall i \in M\forall j \in N$$

What's the rule of writing this dual problem?

Everything is explained here: http://en.wikipedia.org/wiki/Linear_programming#Another_example.


How does the economy "detect" if money is removed from circulation? It is well known that there is a strong connection between introducing more money into an economy and inflation, as well as removing money and deflation.

However, how does the economy as a whole "detect" if money is removed from it?

A similar question would be how it detects whether the supply of money is increased, but that I could understand and explain better as printing new money is regulated, so the government and banks know about the increase and act accordingly. (yes, I know, printing money is not the only way to produce it, but I would guess creating new debt is also regulated or measured in some way)

I could also explain extreme cases, where the amount of money created or destroyed makes up a significant portion of the total amount. If suddenly such a large amount of money appeared, people would start to buy more things, and as the supply of things to buy is not enough, people will pay more money for it to be able to get it by out-bidding others, so prices increase, and we have inflation. Or if a significant portion of the money supply is destroyed, there is an abundance of things to buy, but not enough money to buy things, so prices will have to be lowered otherwise nobody would buy anything, so we have deflation.

However, by what mechanisms does it happen that small (but not negligibly small) amounts of change in the money supply cause small amounts of inflation or deflation?

For example, if one very rich person would sell off some property in cash or take cash out of bank accounts, and just burn or bury that cash without telling anyone, how would the economy "know" it and by what mechanisms would it cause deflation? The amount in this example is not enough for people to consciously realize that money is missing from the economy so they could adapt their spending habits accordingly.

Money is only inflationary in so far as it's moving around. So for there to be an inflationary or deflationary signal, that means that either the amount of money moving around has changed, or its velocity has changed.

And what does it mean to have money "moving around"? It means that it's being used to buy goods or services. And that's how it affects prices - because it affects demand.

So, yes, to elucidate: it's as you say- "if the millionaire wouldn't have destroyed the money but spent it, he could have bought a lot of goods with it, and the sellers of those goods could have bought still more goods with that profit and so on. As now those goods are no longer being bought, the sellers have to slightly reduce prices, so their goods aren't wasted in their warehouses. This means prices will drop a little, and this is how the economy "feels" the difference"


Can value be defined as any increase in consumer or producer surplus? I know that the term Economic Value Added is an internal measure that executives use to measure the financial performance of a company, say, executing a project. However, I'm doing some research in Complex Systems, and I'm interested in the term "value" as it relates to the economic system as a whole.

I came up with a definition of adding "value" as any activity that creates or increases the consumer or producer surplus. I'm now struggling to see how this definition related to the standard definition of Economic Value Added. What are the pros and cons of my definition as against the standard EVA term? Why isn't the definition that I'm using the standard definition, at least in economics?

The standard (finance) definition of EVA is only related to the producer's side of the market. Broadly speaking, it is defined as profits minus the cost of capital.

This is closely related to the concept of producer's surplus in microeconomics which is defined as profits (P) minus fixed costs (FC). In micro-economics, fixed costs most of the time refers to the cost of capital, capital being the fixed factor of production (and labour the variable factor of production).

Producer's surplus (PS) is defined as total revenues (TR) minus variable cost (VC, cost of using labour, in an easy model VK=w*L, with w=wage; L=labourinput). Profits are, of course, total revenues minus total costs (TC), with total cost equalling fixed plus variable costs.

So, it is easy to show from these two definitions that PS really looks like the finance EVA, because, as I said, FC can be seen as the cost of capital, the fixed factor of production.

PS=TO-VC

W=TO-VC-FC

<=> W = PS - FC <=> PS = W + FC

Therefore, the PS (or, for the finance guys: EVA) is only measuring value that is created (surplus) for the producer. Just add consumper surplus (CS) to the PS and you get the total surplus of an exchange. Two remarks:
1) to get the CS you need to measure utility, CS = Utility of the goods bought - price paid for the goods. Unfortunately, this is quite impossible to measure.
2) I use "exchange" since surplus is only created when a seller sells his stuff to a buyer. CS refers to the surplus of total utility derived from the consumption of the goods above the price paid for the goods, PS refers to the benefit of producing (and selling) goods above the situation of not producing. This interpretation is equivalent to PS = TO-VC. When the producer produces and sells his product, he has profits TR-VC-FC. When he doesn't produce, but disposes of the production capacity (the capital), his profits are -FC (no revenues and no variable cost, but having the production capacity at his disposal means he has to bear the cost of capital, the FC). The producer will be producing when is PS is positive, that is, when producing creates a surplus (more profits than if he would not produce).

Notice how I prefer to use surplus (instead of added value) to adress this kind of measure for the value of an activity. This is because, in economics there is the concept of added value, which may be closer to what you're looking for. It is defined as outputs - intermediate inputs (net outputs), and thus measures the value that the producer 'adds' to his inputs when producing. It can be easily measured and is, in fact, GDP.

To sum up, these all the measures discussed:

EVA, the finance equivalent of PS;
total surplus = PS + CS, but with CS being unmeasurable;
GDP, or value added.


Does random sampling cause zero conditional mean? In the lecture notes of my development economics class, it says that
" In the regression model : Yi = β0 + β1Xi + ui,
if Xi is randomly assigned, then Xi
is independent of ui, i.e., E(ui|Xi) = 0, so OLS yields an unbiased estimator of β1."
And the professor says the same thing in his lecture recording.
I really don't understand this statement, because in econometrics (linear regression part)
I thought random sampling and zero conditional mean (E(ui|Xi) = 0) were two separate assumptions which led to the OLS unbiased estimator theory, not one affecting the other.
But the lecture note is saying that random sampling causes zero conditional mean, instead of my
previous notion that the two are separate assumptions which are used together to derive the unbiasedness.
Is the lecture note correct in that random sampling actually causes zero conditional mean? If it is, could anyone explain to me why this is the case instead of them being two separate assumptions as stated in a
standard econometrics textbook?
random sampling causes the sample Xis to be iid but that still doesn't have anything to do with ui, and I thought that was why the additional assumption of zero conditional mean was added to ensure the unbiasedness of the OLS. I would appreciate it if someone could tell me what the lecture note is trying to say here.

The $E(u_i|X_i) = 0$ can hold even without having simple random sample or random assignment. However, random assignment guarantees this will hold (in expectations). A violation of $E(u_i|X_i) \neq 0$ is typically consequence of omitted variable bias. For example, in regression of education on wages reason why $E(u_i|X_i) \neq 0$ can be that experience also affects wages and if people with different education have different experience this will result in $E(u_i|X_i) \neq 0$.
Random assignment of $X$ solves this issue. For example, lets consider $X_i$ to be treatment status $X_i = \{0,1\}$ following Angrist and Pischke Mostly Harmless Econometrics. Now let $Y_{i}$ be potential outcome based on $X_i$ so that we have:
$$ Y_i \begin{cases}
Y_{1i} \text{ if } X_i=1 \\[2ex]
Y_{0i} \text{ if } X_i=0
\end{cases} $$
As a consequence observed outcomes will be given by:
$$Y_{i} = \underbrace{Y_{0i}}_{\beta_0 } + \underbrace{(Y_{1i}-Y_{0i})}_{\beta_1}X_i\tag{1} $$
Now the above in expectation notation implies that:
$$\underbrace{E[Y_i | X_i = 1] - E[Y_i | X_i = 0]}_{\text{observed difference}} = \underbrace{E[Y_{1i} | X_{i} = 1] - E[Y_{0i} | X_i = 0]}_{\text{Average treatment Effect on Treated}} + \underbrace{E[Y_{0i} | X_{i} = 1] - E[Y_{0i} | X_i = 0]}_{\text{Selection Bias}} \tag{2}$$
Presence of selection bias would cause $E(u_i|X_i) \neq 0$.
However, note if we use random assignment of $X_i$ it eliminates the problem as it makes  $X_i$ independent of potential outcomes. If they are independent then $E[Y_{0i} | X_i = 1] = E[Y_{0i} | X_i = 1]$. Substituting this back to the 2 we get:
$$E[Y_i | X_i = 1] - E[Y_i | X_i = 1]= E[Y_{1i} | X_{i} = 1] - E[Y_{0i} | X_i = 1] \\
= E[Y_{1i}- Y_{0i}  | X_{i} = 1] \\ = E[Y_{1i}- Y_{0i}] .$$
This eliminates selection bias and ensures $E(u_i|X_i) = 0$ (although note random assignment is no silver bullet,  there could be various different issues that are still).


Heuristics with time series measured in dollars Suppose the price indexes are adequately measured and there are no problems regarding the data in itself. By my experience, using a sector price index -- for the industry sector that you might be analyzing -- is more adequate than a more general price index for deflating the series. But there could be a situation in which it is more adequate to use a more general price index than a sectoral price index for deflating the variables?

If your analysis is trying to provide predictions for the particular sector at hand in isolation, then one would ask why would you go for more than the index for that sector. However, one then notices that your index is the most comprehensive one possible for your research due to the isolation.

The last point brings one to ask the question of what happens whether the analysis will need a different index if the aforementioned isolation is relaxed. In that case, it is highly likely that you would need to change the benchmark index depending on the purpose. For instance, if a capital owner is trying to find a sector to invest in, it might be in his best interest to see how different sector are doing with respect to each other (or a benchmark) to come up with a better educated decision. An equivalent concern would be subsidies for instance.


How to model best combination of resources I don't have an economics background, I come from CS. I am sorry if this question seems weird.

We want to model the combination of a set of resources to either answer the cost of the model, minimize the cost of the model, or find the cheapest combination of resources.

Precisely, we have a network of computers, with some producing data, and some consuming data, organised in stages of Producers and Consumers: P -> C -> P -> C ... but the network is more like a graph with cycles on the data. We already have the model of this communication paradigm from a CS point of view, we are interested on the Economics side since we have to scale, redesign, and build the system in the future.

What are the existing approaches to model such questions?

It sounds like you could simply use the Cost Minimization Problem:
$$\underset{z_1,...z_N}{min}\sum_{i=1}^N q_iz_i$$
$$s.t.\quad f(z_1,...,z_N)\geq \bar{y}$$
$$z_1,...z_n\geq0$$
Where $z_i$ and $q_i$ are the quantity and price of input $i$, respectively, $\bar{y}$ is some predetermined level of output, and $f(\frac{}{})$ is the production function.

A production function relates physical output of a production process to factors of production. Of course, it seems to me that it may be a challenge to characterize a production function for your case. However, the production function can be any function which satisfies the following:

1) Strict Monotonicity: If $ z'>z$ then $f(z')>f(z)$

2) Quasi-concavity: $V(y)=\{ z:f(x)\geq y\}$ is a convex set

3) $V(y)$ is closed and non-empty

4) $f(z)$ is finite, nonnegative, real valued, and single valued $\forall z\geq 0$

5) $f(z)$ is a $C^2$ function

To be more specific to your case (data is the input and output) the problem reduces to:
$$\underset{d_i}{min}\; d_iq+wl+rk$$
$$s.t.\quad f(d_i)\geq \bar{d_o}$$
$$d_i,l,k\geq 0$$

Where $d_i$ is the data used as input, $d_o$ is the data output, $q$ is the price of input data, $w$ is wage, $l$ is labor hours, $r$ is the rental price of capital and $k$ is quantity of capital.

Again, this type of production process is foreign to me, so I cannot make an informed suggestion regarding specification of the production function, but maybe someone who is more apt on this site may offer a suggestion. I hope this helps!


Local Non-Satiation Proof I have been having trouble with how to go forward with a proof for about three days now. I know the basic structure of the proof, but can't seem to construct it.

Basically, I am trying to do a proof by contradiction for the following:

Say $u: x \rightarrow \mathbb{R}$ has no local maxima. Let $p \in \mathbb{R^l}_{++}$ and $w>0$. Show that if $x^*$ is a solution to the maximization problem:

$$\max_x \ (u(x)) \ \text{s.t.} \ x \in B(p) = [x \in \mathbb{R^l}_{++} : p \cdot x \leq w]$$
then for all $y \in \mathbb{R^l_{++}}$ such that $u(y) \geq u(x^*)$, then it must be that $p \cdot y \geq w$

So I'm supposed to do this proof by contradiction, (suppose we have $y \in \mathbb{R^l_{++}}$ such that $u(y) \geq u(x^*)$, then it must be that $p \cdot y < w$) and use the fact that $u$ doesn't have a local max implies that the function $u$ is locally non-satiated:

$$\forall y \in \mathbb{R^l_{+}, \forall \epsilon > 0, \exists {y'} \in \mathbb{R^l_{+}}} \ \text{s.t.} \ \|y - y'\| < \epsilon \ \text{and} \ y' \succ  y$$

But I've been stuck for a while now. Any help would be appreciated.

Assume for a contradiction that there exists $y$ such that $p\cdot y<w$ and $u(y)\geq u(x^\ast)$. Let
$$\epsilon^\ast= \frac{w-p\cdot y}{ \sum_i p_i}.$$ Then, for all $y'$, if $\ \|y - y'\| < \epsilon^\ast$, we would have $\left|y_i-y_i'\right|<\epsilon^\ast$. Notice that the cost of any bundle $y'$ would satisfy
$$
\begin{eqnarray}
p\cdot y' &=& p_1 y'_1 + p_2 y'_2 + \dots +p_n y'_n\\
&<&p_1 (y_1+\epsilon^\ast) + p_2 (y_2+\epsilon^\ast) + \dots +p_n (y_n+\epsilon^\ast)\\
&=& p\cdot y+\epsilon^\ast\sum_i p_i \\
&=&p\cdot y + \frac{w-p\cdot y}{ \sum_i p_i} \sum_i p_i\\
&=& w
\end{eqnarray}
$$
which shows that for $\epsilon^\ast$, every $y'$ satisfying $\ \|y - y'\| < \epsilon^\ast$ is affordable.

However, since $u$ is locally non satiated, there exists $y^\ast$ with $\ \|y - y^\ast\| < \epsilon^\ast$ such that $y^\ast\succ y$. Since $y^\ast$ is affordable, i.e. $y^\ast\in B(p)$ this leads to a contradiction with the assumption that $x^\ast$ solves the utility maximization problem because transitivity implies that $y^\ast\succ x^\ast.$


Why does Venezuela have such a high Big Mac Index? Venezuela is ranked #2 in the 2014 Big Mac Index at \$7.15 per Big Mac. You can see the graph here.  However, Venezuela is not a very rich country, having a gross domestic product of \$11,789 per capita.

There are at least 20 McDonalds in Caracas, so there is a fairly large supply. With a small gross domestic product the quantity demanded should be smaller by high prices. Even if the market for Big Mac-like products is a monopoly in Venezuela, I still think the price shouldn't be that high.

Another thing that is remarkable is that after 2014, the price had a rapid decrease. Maybe the transformation from a monopoly market to a monopolistic competition market caused it rapid decrease in price in June 2015?

In short the questions:

How could the price of a Big Mac be so high?
What caused it rapid decrease in price?

It is worth noting that Venezuela has had currency controls in place for quite some time now, so the locals cannot get dollars at the world exchange rate.

  On January 2010, the government of Venezuela created a two-tiered official exchange rate system. Imports designated as "non-essential" receive a rate of 4.3 bolivares per dollar, and "essential" goods are exchanged at a rate of 2.6. There is also a third and unofficial exchange rate in the black market valued at around 6.8 bolivares per dollar (March 2010).
  The Venezuelan government decides what "essential" goods qualify for the 2.6 rate. They include imports for sectors related to food, health, education, equipment, and technology; remittances to relatives settled abroad; students' academic expenses abroad; expenses related to health, sport, culture, and scientific investigations; payments to retired and resident pensioners abroad; and currency conversions related to diplomatic activities.

The dollar value of a Big Mac at the official Venezuelan exchange rate is less than \$7.15.
(If my deeming it non-essential is correct it would be around \$2.50.)

Truth be told I am surprised they still have McDonald's restaurants in Venezuela, because long time president Chavez chastized US companies strongly and frequently and I think everyone associates McDonald's with the US.


Microeconomics: Question on Market for Factors I am stuck and confused by a section of what I read in my textbook, could you please help me understand the paragraph?

  If you only vary one means of production and the market for it is competitive you should employ as many units of that means of production as you need to make its Marginal product equal its per-unit cost

  1) The Wage (in terms of good we produce) is  ω = w/px which is the nominal wage (W) divided by the price of the good and it must be the same as the marginal product of Labour for any given level of capital, which means that Money Value of the Marginal Product equals the nominal wage

My confusion stems from how can wage equal wage/price? That seems illogical to me!

Moreover, what does it mean by equals nominal wage? My understanding of Market for Factors was, you hire labour as long as the last labour output is higher than your costs, but I get confused by the term nominal.

Thanks

Nominal means measured in present day money. E.g. someone in South Africa may have a monthly salary of 100,000 South African Rands. You have no idea how much this is unless you are told what real goods can be purchased from this. If a TV costs 10,000 Rands then the real wage of this person would be 10 TVs. Now you see the difference between nominal wages and real wages.

If you reread your question focusing on this part: "The Wage(in terms of good we produce)" I hope all your confusion will go away.


Doing a PhD Economics without background in formal theorems? I could really use some advice. I am a new PhD student in Economics and Finance. My background is in economics, and while I did mathematics in my Master's course, it was more data-analysis based rather than formal proofs. So, for instance, I have become quite adept at analysing data in SPSS, STATA, R etc, but struggle when it comes to being given a formal problem and then having to prove using formal means; e.g. proving that a particular sequence is Cauchy or Martingale, etc.

What are the best ways to become better at formal theorem-proofs? I know that practice is the most obvious answer, but is there a base from which I should be starting? e.g. are there certain "bread-and-butter" proofs in mathematical economics such as law of large numbers, central limit theorem etc, that being able to prove would make it easier to prove others?

Would appreciate some advice on how to get better at the theorem-proof method, or even some textbooks/resources that would help with this, given that my background in this area is very limited.

I did my undergraduate double majoring in discrete mathematics and economics. I went through the grad micro sequence, the math for economists grad course, and the game theory field course, though. So I have a good idea of what the courses are like on both ends. (Doing the grad school route now, but that's additional commentary with which I won't bore everyone).

Traditionally, coursework like abstract Linear Algebra and Number Theory have been used by math departments to teach proof writing. In the last 15-20 years, an Intro to Proofs class has been added. Number theory doesn't come up appreciably in economics. So I'd punt that. But a solid course in linear algebra is so important. And if you have a strong data science background, then you should be very comfortable with the material, though perhaps not the formalization.

Real analysis is the maturity sequence in the math departments. Rudin is the standard textbook. If you take it, expect to be putting in a solid 20 hours per week, but you will be razor sharp by the end of it. Real analysis is usually a senior or introductory grad sequence. There is often times an introductory analysis class that uses a text like Abbott. This introductory class is still something you shouldn't take lightly, but it should be much easier and sufficient for a lot of what you'll do in economics. If you are working in stochastic processes, martingales, etc., then measure is likely to come up, and so you should take the Rudin sequence at some point and follow it up with measure theory (which is a very tough grad course).

The math for economists course compresses much of these courses (usually not measure) into a semester. When I took it, we trucked through the abstract linear algebra semester-long course in three weeks. It was also an ugly presentation, but that's what happens when you push for speed rather than elegance. And this is one more reason why you should take these courses in the math department.

Combinatorics and graph theory is also becoming increasingly important in economics, with regards to networks. So a course in graph theory wouldn't hurt either. Graph theory proofs are picky in the sense that you have to be very precise in saying "because it looks like it." This is different than analysis and algebra proofs. Graph theory is my particular area, so I can definitely speak to it.

A course in analysis of algorithms or theory of computation wouldn't be a bad thing to have either, with complexity becoming increasingly important in economics (algorithmic mechanism design, algorithmic game theory, strategic network formation, showing computing equilibria is computationally difficult for certain classes of games, etc.).

To sum it up, I'd strongly recommend:
Abstract Linear Algebra
Intro to Real Analysis

I'd recommend you consider:
Real Analysis (Rudin)
Measure Theory

Nice to Have courses:
Graph Theory
Combinatorics
Theory of Computation
Analysis of Algorithms

Let me know if you want book recommendations for any of these courses.


The influence of restricted type space on incentive compatible mechanism design? When we are to design an IC mechanism for an auction, we usually assume that each agent $i$'s type $\theta_i$ is draw from its type space $\Theta_i$, which contains all the possible types of $i$. For instance in a single item auction, we may assume that $\theta_i \in \mathcal{R}^+$, i.e., $\Theta_i = \mathcal{R}^+$.
In some scenario, prior information about the type space can be acquired by the auctioneer by some means. For example, the auctioneer may confirm that $\theta_i \le M$ before designing any mechanism. That is, the maximum money each agent can pay is no more than $M$. The auctioneer may obtain the information from some statistic query in data market, if we have to explain how she gets it. Then we can validly assume that $\Theta_i$ is $[0, M]$ instead of $\mathcal{R}^+$. Suppose some agent can only benifit himself by misreporting some $\theta_i^{\prime} > M$. Then we have no worry about truthfulness because now $\Theta_i = [0, M]$. Note that agents would not report higher than $M$ because they are informed of the auctioneer's access to the statistics(privacy issue is not considered here).
The above example may not be well constructed, however I'm considering similar setting where  some constraints on agents' type space may make truthful mechanisms easier to get, or even make nontruthful mechanism truthful. I failed to find out any literature on this topic. I appreciate if someone can provide guidance on where to look for such research and whether this is valid as a research point.

The type space is the support of the belief about the types. It seems a bit weird to allow a mechanism designer to arbitrarily restrict their belief support. However, you are right: if the designer can find out more about the private types by acquiring information, this would intuitively allow them to reduce information rent, because it makes the type "less private." Of course it would be best to fully unveil types such that no incentive for revelation has to be provided and full surplus extraction becomes possible. In your constructed example it may not be helpful to exclude very high types $>M$ as more profitable deviations are often found in closer vicinity to the real type. That depends on the problem at hand of course.
I can't see clearly where you want to go with this. So let me suggest something else in a similar spirit: mechanism design with evidence. Here, agents can make their private information hard by providing evidence that their true type belongs to some subset of the type space. See, e.g., the work by Ben-Porath, Dekel & Lipman (AER 2014, Ecta 2019 and more), Bull and Watson (GEB 2007), Hagenbach, Koessler & Perez-Richet (Ecta 2014) and others.


Are there multiple equilibria in the second price auction? Suppose that $n\geq 2$ bidders compete in a second price auction. Each bidder $i$ knows their own valuation $v_i$, but only knows the distribution generating the valuations of the other players. Valuations are independently, continuously and symmetrically distributed. Define a (pure) strategy of a player as a function mapping from each valuation that they might have to a bid $b$; and define a (pure strategy) equilibrium as a set of (pure) strategies, one for each player, such that each player's strategy maximises their expected payoff given the strategies of the other players.

In this context, it is well known that it is weakly dominant for each player to bid their valuation, i.e. set $b(v_i) = v_i$. However, are there are other (pure strategy) equilibria of this game?

One point that perhaps should be clarified: in some discussion of second price auctions that I have seen, people define each player's strategy as their bid (a number) not their bidding functions. Here I am interested in the game obtained by defining strategies as functions.

Edit: in response to some helpful examples from @Giskard, we may wish to restrict attention to equilibria that (i) involve deviations from truthful bidding that occur with positive measure (ii) survive natural refinements that we might want to apply.

Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies
$$
b_1(v_1) = v_1
$$
and
$$
b_2(v_2) =
\left\{\begin{array}{cc}
v_2 & \text{ if } v_2 < 1 \\
5 & \text{ if } v_2 = 1.
\end{array}\right.
$$
Another, slightly more annoying equilibrium for $v_1,v_2 \in [0,1]$:
$$
\begin{align*}
b_1(v_1) & = 0 \\
b_2(v_2) & = 2.
\end{align*}
$$

EDIT: The truth-telling bidding strategies are weakly dominant strategies. If we only allow trembling hand perfect equilibria, players assign positive probabilities (in this case densities) to all truncated strategy profiles. In such a setting, the weakly dominant strategies become strictly dominant. (Because of sequential rationality even zero measure deviations such as my first example are eliminated.) In a game where all players have strictly dominant strategies, there is only one equilibrium.