Has an increase in housing supply in popular cities including Amsterdam led to an increase in house prices? Common economic thinking dictates that an increase in price should lead to an increase in supply, and that an increase in supply should then lead to a decrease in price.  According to Maarten van Poelgeest, former alderman of Amsterdam, the opposite is happening in Amsterdam and other popular cities.  He is quoted in an article in De Groene Amsterdammer as saying:

  ‘Hoe meer er gebouwd wordt, hoe aantrekkelijker de stad wordt. Er komen meer voorzieningen, er komt meer reuring. Nieuw aanbod maakt de vraag dus juist weer groter – en de huizenprijzen hoger.’

Which means:

  ’The more there is built, the more attractive the city becomes. There will be more facilities, there is more bustle. New supply therefore makes the demand even larger – and house prices higher.’

The reasoning by Maarten van Poelgeest appears plausible, but is there any evidence that this is actually happening?  Is there evidence that an increase in housing supply leads to an increase in house prices, due to housing in the area becoming more attractive?

I'm looking for an answer based on empirical evidence from trends in the past up to the present, in order to present the claim/hypothesis by Maarten van Poelgeest up to the best available measurements.

NB: I am emphatically not asking just for a correlation.  A correlation between increase in supply and increase in price can easily be explained by the reverse of the causation I'm asking for here.  I'm aware that causation is harder to provide evidence for but it should not be impossible.

Let's just say that economists don't really agree with this thesis, and most city-level empirical evidence points in the opposite direction. E.g.

  We ultimately conclude, from both theory and empirical evidence, that adding new homes moderates price increases and therefore makes housing more affordable to low- and moderate-income families.

  [...]

  A few studies use panel data and find that the imposition of more stringent land-use controls
  leads to lower supply and higher prices. Jackson (2016) uses longitudinal data from California’s
  cities to assess the effect a city’s adoption of additional land-use regulations has on the number of
  new construction permits issued, and finds that each additional land-use regulation adopted
  reduced multifamily and single-family permits by an average of more than 6% and 3%, respectively,
  and that regulations reducing allowable density had even larger effects. Zabel and Dalton
  (2011) use longitudinal data from localities in Massachusetts and find that increases in minimum lot
  sizes are followed by significant increases in prices. Looking at longitudinal data on municipalities
  in the Boston (Massachusetts) metropolitan area, Glaeser and Ward (2009) find that the adoption of
  stricter local regulations leads to higher house prices, but the coefficient falls in magnitude and
  loses significance once they control for population demographics. They point out that this is
  expected, if homes in other jurisdictions are seen as perfect substitutes. Thus, whereas supply
  restrictions may increase prices in a market as a whole, they may not increase them disproportionately
  in the particular locality where they are imposed due to spillover effects across jurisdictions.

  Several other researchers use instrumental variables to try to more clearly assess the causal
  effects regulatory restrictions have on housing supply and prices. Ihlanfeldt (2007) uses such an
  approach to study regulation in localities in Florida and finds that predicted regulations significantly
  increase the price of single-family homes. Saks (2008) uses instrumental variables and shows
  that increases in labor demand lead to less residential construction and larger increases in housing
  prices in metropolitan areas with more restrictive housing supply. Hilber and Vermeulen (2016)
  show that changes in demand lead to increases in local house prices rather than increases in
  supply in municipalities in England with greater regulatory restrictions, measured by the refusal
  rate of proposed residential projects and the number of project approvals delayed more than
  13 weeks.

  In sum, the preponderance of the evidence shows that restricting supply increases housing
  prices and that adding supply would help to make housing more affordable. [...] although it is surely true that land is constrained, especially in certain
  markets (Saiz, 2010), land can be used more intensively to allow for more housing. The limits on the
  land with which housing is bundled make housing different from many goods, but the difference is
  one of degree: the supply of housing can and does increase even in constrained markets, and
  prices should generally fall in response (see the review by Dipasquale, 1999; Mayer & Somerville,
  2000).

  [...] most housing filters down, or loses
  value as it ages, representing new supply in submarkets at lower price points. [...] Indeed, recent research shows that filtering
  was the primary source for additions to the affordable rental stock between 2003 and 2013,
  whereas new construction was the largest contributor for the higher priced rentals, and tenure
  conversion was the largest source for moderately priced rentals (Joint Center for Housing, 2015, fig.
  14). Further, Weicher, Eggers, and Moumen (2016) report that 23.4% of the rental units that were
  affordable to very low-income renters in the United States in 2013 had filtered down from higher
  rent categories in 1985. Another 21.8% were conversions from formerly owner-occupied homes or
  seasonal rentals.9 Most of the higher priced rental units that filtered down to become affordable in
  2013 were moderate-rent units in 1985, but 15% of those that filtered down were high-rent units in
  1985. Note that filtering occurs over a shorter time frame too; among affordable units in 2013,
  19% had been higher rent units as recently as 2005.

  Recent research analyzing the incomes of successive occupants of homes also suggests substantial
  downward filtering, particularly of the rental stock due to tenure conversion; as the owner-occupied
  stock ages, a portion converts to rental (Rosenthal, 2014). Rosenthal also finds, however,
  that filtering rates are considerably lower in areas with high house price inflation, although
  downward filtering still occurs.

  In short, new construction is crucial for keeping housing affordable, even in markets where
  much of the new construction is itself high-end housing that most people can’t afford. A lack of
  supply to meet demand at the high end affects prices across submarkets and makes housing less
  affordable to residents in lower-cost submarkets.

Finally getting to the discussion of the precise thesis you ask about:

  Some skeptics argue that even if additional supply could help make housing more affordable in the
  short run, it won’t in the long run because the additional supply will induce more demand, especially
  among buyers or renters wealthier than the existing residents in the neighborhood (Redmond, 2015).
  The claim is analogous to the argument that building more highways will not reduce congestion
  because the lower cost of travel will simply cause more people to drive or to take that particular route
  (Gorham, 2009). In this case, the argument is that by making the jurisdiction more affordable, adding
  housing supply will attract new demand—both from current residents who would otherwise leave,
  and from people living elsewhere who will now choose to move to the jurisdiction. Further, the argument goes, lower rents and prices may also induce latent demand—people who are living with
  roommates or family members may choose to form their own households (Ellen & O’Flaherty, 2007) or
  people may choose to invest in pied-à-terres in a city. That additional demand will drive prices back up
  until supply can again respond, causing housing to be more affordable, at best, only cyclically,
  according to the argument, and increasing the density of the jurisdiction, with the attendant costs
  of congestion.

  Although building additional highways does appear to induce more demand (Duranton &
  Turner, 2011), in the case of housing, additional demand is unlikely to completely offset the new
  supply. Such an offset requires demand curves to be perfectly elastic—or, in other words, it
  assumes that neighborhoods and jurisdictions are perfect substitutes and that there are no
  constraints on the ability and willingness of households to move. That is unrealistic. Moving
  homes is not like driving a few extra miles (Lewyn, 2016), and costs associated with moving may be
  high. Any additional demand induced by new housing is limited by personal and economic
  constraints on the ability and willingness of households to move, restrictions on immigration, and
  uncertainty and other factors that might inhibit renters and buyers from renting or buying in the
  market in which housing supply increases. Indeed, mobility rates have fallen sharply over the past
  several decades, and although the reasons for the decline are being debated, the decline reveals
  significant constraints on the ability and willingness to move.

  Thus, in the long run, whereas some additional households may be drawn from outside (or from
  within the city) to buy or rent homes as supply increases, it is highly unlikely that prices will end up
  at the same level that they would have reached absent any new supply. Finally, as noted above, the
  empirical evidence shows that allowing more supply leads to lower housing prices; if adding
  supply induced sufficient additional demand to offset the increased supply, the studies would
  not find an association between supply and prices.

It can happen indeed in some very limited circumstances, and with localized [e.g. neighborhood] effect, which is probably on what this faulty generalization is based on:

  Many renters in neighborhoods where market-rate housing is proposed express concern that the
  construction of new housing will actually make their affordability problems worse by raising rents or
  house prices, fueling gentrification, and potentially displacing existing residents (Atta-Mensah, 2017;
  Savitch-Lew, 2017). Hankinson (2017) theorizes that renters’ opposition to local additions to supply is
  driven by such worries; he argues that it is plausible that the construction of an attractive new building
  will increase prices locally (by improving the physical landscape, bringing new amenities to the
  neighborhood, and signaling that the neighborhood is improving), even as it reduces them citywide.

  Testing this proposition empirically is quite challenging, given that developers will naturally be
  attracted to areas where prices and rents are rising. There is evidence that improvements to
  blighted housing can, in some circumstances, increase surrounding property values, even when
  the new or improved housing is subsidized, low-income housing (Diamond & McQuade, 2016;
  Schwartz, Ellen, Voicu, & Schill, 2006). The new housing studied, however, typically replaced
  vacant, abandoned buildings and littered vacant lots, in essence removing a disamenity. [...]

  There is little empirical evidence about the net effect new market-rate housing has on the prices or
  rents of nearby homes, and what exists may not be causal. One recent study examines the effect of
  market-rate single-family homes newly constructed on infill sites, and finds that newly constructed
  single-family homes can have positive impacts on the sales price of other single-family homes nearby,
  but the effect varies with context (Zahirovich-Herbert & Gibler, 2014). A study of multifamily high-rise
  infill developments in Singapore found positive price effects on nearby houses (Ooi & Le, 2013), as did
  a study of single multistory apartment buildings constructed in Helsinki (Kurvinen & Vihola, 2016).
  These studies all consider property values and not rents, and none is able to prove a causal relationship
  given that market-rate developments aim to target neighborhoods where they expect property
  values to improve. Unfortunately, we found no study examining impacts on rents, although one study
  by the California Legislative Analyst Office concluded that additional market-rate construction is linked
  to lower displacement rates (Taylor, 2016). Examining low-income neighborhoods in the Bay Area
  between 2000 and 2013, these researchers found that the production of market-rate housing was
  associated with a lower probability that low-income residents in the neighborhood would experience
  displacement. Although a singular study, the findings suggest that for neighborhoods in high-demand
  cities, blocking market-rate construction may place greater pressures on the existing stock.

  In short, although it is clear that the construction of new homes will moderate price and rent
  increases citywide, neither theory nor empirical evidence provides clear guidance about when
  localized spillover effects might occur and when they might actually cause an increase in the prices
  and rents of immediately surrounding homes.

There's a more recent study on the core issue of the question, not covered in the above review:

  Preliminary results using a spatial difference-in-differences approach suggest that any induced demand effects are overwhelmed by the effect of increased supply. In neighborhoods where new apartment complexes were completed between 2014-2016, rents in existing units near the new apartments declined relative to neighborhoods that did not see new construction until 2018. Changes in in-migration appear to drive this result. Although the total number of migrants from high-income neighborhoods to the new construction neighborhoods increases after the new units are completed, the number of high-income arrivals to previously existing units actually decreases, as the new units absorb a substantial portion of these households. On the whole, our results suggest that—on average and in the short-run—new construction lowers rents in gentrifying neighborhoods.

See also a blog discussing it (which also does a good job at debunking the roads analogy); in summary, the gist of the latter study is in this chart:

  This chart shows that rental prices for apartments close to the new building fell relative to the prices for apartments located further away. The dashed “before” line has a negative slope, suggesting that prices declined the further you got from the site of the new building.  The solid “after” line has a positive slope (prices increase the further you get from the new building).  Overall, prices are higher (the solid line is above the dashed line), but prices actually went down next to the new building, and increased far less than in the area further away from the new building.

  These data are a strong challenge to the induced demand theory.  If a new building made an area more attractive, one would expect the largest effect in the area very near the building. But, consistent with the traditional “more supply reduces rents” view, the addition of more units in an area seems to have depressed rents (or at least rent increases) compared to buildings in the surrounding area.

In other words, even the theory that prices might increase locally in the neighborhood of a new building doesn't hold up, on average, to a difference-in-differences test. Without a difference-in-differences approach one can of course draw the wrong conclusion since overall demand in a city may be increasing faster than the overall supply.

Even more recently, another paper on NYC (not yet published in a peer-reviewed venue, but was mentioned in a recent NYT article):

  I find that for every 10% increase in the housing stock within a 500-foot buffer,
  residential rents decrease by 1%. The rent reduction is caused by the completion of new
  high-rises rather than their approval. Across neighborhoods, the impact is smaller in more
  central areas, presumably due to more elastic demand. Within neighborhoods, the impact
  is smaller for lower-rent buildings. Finally, the negative impact appears to be driven by
  supply effects rather than dis-amenity effects, like changes in neighborhood physical features,
  blocked views, or shadows. New housing units alleviate the growing demand for existing
  housing units and moderate the rapid growth of residential rents.

  Residential property sales prices also decrease when new high-rises within 500 feet
  are completed. [...] Because 99% of new high-rises are condos and rental
  buildings, they do not significantly affect sales prices of co-ops and 1-5 family homes. The negative impact is bigger for closer substitutes, further confirming the negative impact is due to supply effects.

  To address the hypothesis about amenity effects, I find new high-rises and their
  high-income tenants attract new full-service restaurants, cafes, and coffee shops. These consumption amenities likely make neighborhoods more attractive and potentially increase rents
  and sales prices (Couture and Handbury, 2017). However, the amenity effect is dominated by the supply effect, given that rents and sales prices still fall on net.

As was mentioned in a OP's comment below, living in a big city vs a small town (and vs the prairie) is an amenity effect. However, there seems to be a limit to this effect after which more housing supply lowers the (rental) price.

The amenity effect has been studied for decades but alas often in isolation, most prevalently perhaps with the the Roback model, which basically adapted Rosen's more general spatial equilibrium model to "nontraded" goods like housing. ("Nontraded" because Roback only considered effects on rents.) Using this model, one essentially gets (rental) price-amenity gradients. Roback assumed that "the cost of changing residences is zero" (i.e. costless migration). Modelling "partially open" cities, i.e. with non-zero (and non-infinite) relocation costs across its boundaries is a relatively more recent endeavour. Likewise for considering the combined effect of housing supply and amenities on rents. As one such theoretical work notes

  as housing is supplied more elastically [...] the rent effect of amenities approaches zero.


Is it better for low-income people to live in the countryside where it's cheap or in the city where there are more services? Supposing someone is struggling to pay the rent in an urban area, and they don't have anything tying them down to any one place, would it make sense to move to a rural area where the rent is much lower? Or are cities generally better for low-income people because of the services like food banks, etc.?

It depends not just on the cost of living, but on the available means of income (wages) relative to each location.  Rural areas may have cheaper rent, but they often have fewer employment opportunities.  The right balance will often depend on the skills/qualifications of the individual.


How startup early valuation influence later funding stages? Let's suppose that a startup raise his seed round with a 100k funding for 10% equity. The startup has a pre-money valuation of 1M.

Let's say that the startup, after 1 year, raise a Series A round. Does the valuation of the startup in the seed round has any impact on the pre-money valuation at the series A?

My understanding is that in principle there is no correlation. Am I right? Is there perhaps a sort of psychological effect by which it's not feasible to value the startup 5, 10 or 100 times more than the previous round? (the time scale of 1 year is arbitrary here)

No why would it? The valuation should solely be based on the future expectations of the company. Perhaps during this one year, the startup has taken on 100x more customers than it had before, or earning 200 times more, and seemingly have an even brighter future than it did one year ago. Why would the previous valuation affect the new one?

This is at least true for rational investors.


Real world application of IS-LM model I learnt IS-LM model but I got curious whether it is applicable to the real world. I mean it's just an model, so I doubt that the realtionship between Y and r actually exists. Are there any real life examples that prove the model? I want to know if something like a recent world event or fiscal and monetary policies in sone countries actually affects output and interest rate.

There is a huge discussion about structural Macroeconomics models - especially on their theoretical and practical relevance - with IS-LM being on the forefront of the battle some 40+ years ago. Many people simply reject LM existence. Advanced level textbooks ignore this approach for two decades. However it ironically dominates introductory courses.
 Here is a nice discussion from Bloomberg that claims that, perhaps, the most instructive paper on the matter was published in QJE back in 1992 - link. (Full text can be downloaded elsewhere.)
From personal experience I could add that IS-LM is a great tool to build initial schematic intuition about the issue in question even though its microfoundations are not clear at all (put it another way, we have no idea why does the model work, but it does).


"Printing" money to build housing If the government prints money to build housing for cheap rental.  Is that a good idea? Or will it damage the economy causing inflation?

Rent will generate income.

Low rent will give poorer more spending money.

This would be government housing, rental only.  Not for sale.

This is looking at the UK, based on the idea of one of the opposition leader candidate's idea.

This is a partial answer, in the sense that it addresses the case for state-sponsored house building, rather than the question of how that should be financed. In general, any means by which the state might finance spending is likely to introduce some kind of distortion into the economy. The question then faced is whether that spending generates benefits sufficient to offset the cost of the distortion.

I agree with the general sentiment of others that if there is a well-functioning, uninhibited market for housing then any intervention in that market (such as an artificial increase in supply) is likely to reduce overall efficiency and, in particular, lead to the misallocation of resources within the economy more broadly. @suriv outlined the intuition for this.

However, if there are distortions that prevent the housing market from functioning efficiently then a state-sponsored increase in housing supply can be efficiency enhancing. This is a simple application of the more general "Theory of the Second Best", which holds that if there is pre-existing distortion in the economy then introducing another distortion can sometimes make things better, not worse.
 There are various reasons why the housing market might not function efficiently. Here are a couple:

Planning laws are sufficiently restrictive so as to reduce the market's ability to supply housing. For example, if planning permission is hard to obtain then developers who have planning permission have a kind of monopoly power over the ability to build houses and are likely to ration supply in order to increase profits. N.B If this is really the problem, then reform of the planning system is likely to be more efficient than a direct housing market intervention. But such reform might also be politically problematic.
Property builders are extremely vulnerable to economic downturns because housing demand is pro-cyclical. They are therefore unwilling to commit to risky, large-scale construction programmes that have long time-horizons.
Speculation on rising property prices induces people to hold undeveloped land or un(der)occupied properties in the hope of selling later when the price is higher.

All of these factors seem to be important considerations in the UK market (which is, I guess, the example that motivated your question). Since we are talking about state-sponsored house building, here is a graph that shows how much of the slack private-sector housebuilders in the UK took-up when the state withdrew from house building in the 1980s:

The answer is "basically none".

But does that mean that the market is now failing to supply enough housing, or just that there was previously an inefficient excess of supply that the market has now corrected? A figure that, I think, is broadly indicative of the fact that the UK housing marketing is currently not functioning well comes from this UK government report. It states that the UK is currently building houses at a rate of 135,000/year, but creating new households at a rate of 240,000/year. The 240,000 figure does not include either backlog from undersupply in past years, or demand for housing from foreign speculators (the FT reports that 75% of new homes in London are bought by foreigners, many of whom are speculators or will otherwise not occupy the property full-time). The UK has never in its history managed to build 240,000 homes per year except at times when the state accounted for close to half of all house building.

As somewhat of an aside, the effects of this long-term failure to meet demand upon the price of UK housing has been significant. Here is a comparison of various national house price indices from The Economist:

If one believes that such price growth is the consequence of a distortion in the market then it has important knock-on consequences for other aspects of economic efficiency: it encourages speculation and high levels of indebtedness, distorts investment decisions, and induces internationally mobile workers to leave the UK. As regional price disparities are magnified by rapid price growth, internal migration is also potentially distorted. If we care about equality then this kind of price growth also implies a growing wealth gap between property owners and non-owners.


How do tariffs work? When a foreign company wants to export their goods to the U.S. they may encounter a tariff (e.g. steel). The tariff is imposed by the U.S. government at the port of entry and is a form of tax revenue. However, does the exporter's revenue take a hit or do they simply charge the importer more to take the cost of the tariff into account? Does it just depend on whatever agreement the exporter/importer come to?

Assuming a competitive global market,

If an importer would by steel in the global market in global prices, he will have to absorb the tax cost at importing, then roll it to its customers at home
The exporter will not absorb the tax, since it can sell to other countries without it.

If the importer's country is big enough market-wise, then:
some tax absorption on the exporter's side might take place since the exporter will lose large sales if no absorption is made,
And the importer will not roll all the tax to its customers since competing importers, might receive the same tax-absorption sharing with exporters (theory dictates it, all are equal in a competition),
In this case, a lower price from competitors will lead to a loss of market share


hicksian demand of perfect complements As the title states, I want to know how to derive the hicksian demand of perfect complements $\text{min} \, \{x1,x2\}$.  Thanks in advance.

Also, no price is given, or budget.  my main question is how do i do this considering there really is no utility function?

Hicksian demand is the consumption bundle that minimizes the expenditure of the consumer subject to the constraint that he attains some target level of satisfaction in equilibrium. In the problem, the expenditure on any bundle $(x, y)$ is given by $p_Xx + p_Yy$ and the target level of satisfaction is $\mu$. Given that the utility function is $u(x, y) = \min(x, y)$, the expenditure minimization problem is formulated as:

\begin{eqnarray*} \min_{x, y} \ \ & p_Xx+p_Yy \\ \text{s.t.} \ \ &  \min(x, y) \geq \mu\end{eqnarray*}

Solution to this problem is a function known as Hicksian demand function :

$x^h (p_X, p_Y, \mu) = \mu $

$y^h (p_X, p_Y, \mu) = \mu $


From an economics perspective, what are the ramifications of a currency with fixed money supply? I'm thinking specifically of bitcoins.

What are the pros and cons of having a fixed number of coins, as opposed to more "normal" currencies?  Would the currency have no inflation?

FooBar is quite right that unless you expect GDP growth to stop, fixed nominal supply currencies will lead to deflation.

A moderate degree of currency inflation serves a number of useful functions in the economy. The most obvious are:

It induces people to spend their money before it loses its value. In
a deflationary environment there is an incentive to put money under
your mattress and spend it in a year when it has greater purchasing
power. If everbody does this then the lack of demand will lead to a
decrease in overall economic activity (i.e. a recession).
It provides a weapon against downward nominal rigidities. For example, workers are generally reluctant to accept a nominal pay cut, even if market conditions are such that the current wage is above the equilibrium level. Inflation means that their employer can simply increase wages at less than the inflation rate so that the real wage is decreasing.
It erodes the real value of nominally denominated debt. Now, this is obviously only a pseudo-advantage because (whilst it benefits debtors) it harms creditors. However, this kind of erosion of debt may be desirable if national economic stability is threatened by high debt levels. Also, since debtors are usually poorer on average than creditors, it can reduce inequality, which may be a normative objective for the government.

Without inflation you miss out on these benefits. The first benefit might not seem like a big deal if you think you can simply set the rate of inflation at zero percent. But it is very hard to hold inflation constant at some target level so attempting to hit zero inflation will almost certainly result in occasional lapses into deflation, with the attendant negative economic consequences.


Difference between double auction and continuous double auction? What is the difference between double auction and continuous double auction?  Sometimes I feel that they were types of two-sided auction but sometimes I feel as if they were two separate concepts.

So, if a distinction is made, as continuous double auctions are usually just called double auctions, then the difference has to do with frequency.  It is easier to have an example.

The New York Stock Exchange is an example of a continuous double auction.  Within its trading hours, you can bid in either direction as much or as often as you want and so can everyone else.

The best example of an ordinary double auction was the Arizona Stock Exchange which closed in 2001.  It only permitted two bids per day, at the open of US trading and at the close.  It sought to solve an institutional liquidity problem.  Certain types of institutions are at risk if they cannot clear their orders at or as near to as possible a particular moment in time.  Mutual funds, for example, usually only have one valuation per day and if there need to be a lot of transactions from redemption or purchase activity, then the least risk is to have the trade near that time.  Most are at the open or close.

The Arizona exchange found the single price that cleared the largest number of trades.  You paid a fee to participate.  There was no bid-ask spread.  So possibly hundreds of customers would place their limit orders at the same time as buyers or sellers.  What they did in practice was create single supply curve and a single demand curve and the price paid/received was the intersection of the two curves.


UK Government Coronavirus Job Retention Scheme Funding During the Covid-19 outbreak the UK government is offering to pay upto 80% or a max of £2500 per month to prevent employers laying off workers;

  Furloughed workers
  If you and your employer both agree, your employer might be able to keep you on the payroll if they’re unable to operate or have no work for you to do because of coronavirus (COVID-19). This is known as being ‘on furlough’.

  Your employer could pay 80% of your regular wages through the Coronavirus Job Retention Scheme, up to a monthly cap of £2,500.

In some instances wealthy individuals and companies are using this government fund to pay employees instead of taking the burden on personally/commercially which would be entirely feasible for them. This has created anger towards these people/companies.

However, as I understand it, the money that the government pays out will no doubt be recouped by the government at some point through an increase in personal income tax/VAT(Sales Tax)/Corporation Tax or a combination of all three. In which case the government is essentially providing a loan with repayment an expectation. Is this correct? in which case it would be perfectly reasonable for employers to use the Coronavirus Job Retention Scheme even if they could afford to make the payments themselves rather than laying off the workers.

It is not correct to say that the Coronavirus Job Retention Scheme is essentially providing loans.

A little background.  The UK government currently has two major schemes to support businesses and through them their employees: the Coronavirus Business Interruption Loan Scheme for small / medium businesses together with a similar scheme for large businesses, and the Coronavirus Job Retention Scheme.

As their names suggest, the former is a loan scheme, the latter isn't.  As you imply, the cost to the government of the Coronavirus Job Retention Scheme will have to be funded somehow, probably initially by extra government borrowing, but perhaps eventually by increased taxation. Depending on the form of taxation, some of the burden of that extra taxation might fall on the businesses and/or employees who benefited from the Scheme.

However, there is no reason to expect that there would be any close match, at the level of individual businesses, between their receipts from the Scheme and any extra taxation they might have to bear.  Corporation Tax, for example, is subject to complex rules which bear little relation to eligibility for receipts under the Scheme.  It would not be correct therefore to regard the Scheme as offering loans with an expectation of repayment.


Consumer Surplus question The demand for rail travel is $Q^d = 600 - 2P$ where quantity is thousands of train journeys per quarter and $P$ is in £ per journey. How much would the consumer surplus change if rising cost of electricity led the train companies to raise price from £70 to £100.

Calculating the choke price, $600 = 2P$. Thefore $P = 300$.

At £70, $Q^d = 460$. Therefore Consumer surplus is $(460-300)/2*460 = £36,800$

At £100, $Q^d = 400$. Therefore consumer surplus is $(400-300)/2*400 = £20,000$

Difference is $£16,800$.

I've got the answer wrong, but i'm sure where, any help is appreciated.

Image courtesy http://economicsonline.co.uk/

Consumer surplus is the sum (integral) of differences between the price each consumer would have payed and the price they got to pay. You need to find out the area of the green zone on the above graph, in the case of your model.


AGV mechanism and Individual rationality I have the following question. I consider AGV mechanism which is as follows:
$u_i(x,\theta_i) = v_i(k,\theta_i) + t_i $ is an utility function where $x = (k,t_1,...,t_n)$ vector of alternatives. There are N agents. We choose $k^*$ such that $\sum v_i(k^*(\theta) ,\theta_i) \ge \sum v_i(k^*,\theta_i) $ and take
$$t_i(\theta) = E_{\tilde{\theta}_{-i}}\Big[ \sum_{i \neq j}v_j(k^*(\theta_i,\tilde{\theta}_{-i}),\tilde{\theta}_j)\Big] + h_i(\theta_{-i}) = \xi_i(\theta_i) + h_i(\theta_{-i}). $$
Also we define
$$h_i(\theta_{-i}) = - \frac{1}{N-1} \sum_{i\neq j}\xi_j(\theta_j)$$

Now I want to check whether is it true or not that this mechanism satisfies ex post individual rationality. There are some sources claiming that AGV mechanism violates ex post IR but satisfies ex ante IR.

Ex post IR is the following property:
Mechanism $\phi$ satisfies ex post IR iff $\forall \ \theta, \forall \ i $ we have: $v_i(k;\theta_i)+t_i\ge 0$.

So here we have:
$$v_i(k;\theta_i) + \xi_i(\theta_i) \ge \frac{1}{N-1}\sum_{j\neq i} \xi_j(\theta_j)$$
In general there are now reasons for this inequality to hold. But what might be an example that this inequality is violated?

The simplest example I could cook up. I might have messed up somewhere.

Consider the provision of a public good with two agents, $n=2$. Let the type space be such that $\theta \in \{0,1\}$, and both types are equally likely. Either the good is provided, $k=1$, and total cost $c=2/3$ occurs, or the good is not provided, $k=0$, which is costless.
Let
$$v_i (k; \theta_i) = k  (\theta_i - \frac{c}{2}).$$
That is, either the good is not provided and nobody pays or it is provided and both share the cost. Of course transfers can be used to compensate.
It is efficient to provide if at least one agent values the good, i.e., has a type of 1. AGV leads to this efficient rule $k^* (1,1)=k^* (0,1)=k^* (1,0)=1$ and $k^* (0,0)=0$.

Then,
$$\xi_i (1) = \frac{1}{2} \left( v_j(k^*(1,0);0) + v_j(k^*(1,1);1) \right)
= \frac{1}{2} (0+1 - c) = \frac{1}{6},$$
$$\xi_i (0) = \frac{1}{2} \left( v_j(k^*(0,0);0) + v_j(k^*(0,1);1) \right)
= \frac{1}{2} (0+1- c/2) = \frac{1}{3},$$

Now, suppose $(\theta_i,\theta_j)=(0,1)$. Then,
$$v_i(1;0)+\xi_i(0) = 0 - \frac{1}{3}  + \frac{1}{3}= 0 < \xi_j (1) = \frac{1}{6}.$$
So, we don't have ex-post IC in that case.


Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations There is one agent with utility function given by:

\begin{equation}
U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1}
\end{equation}

With budget constraint:

\begin{equation}
c_{t} + k_{t+1}   = (1+r_t)k_{t} + l_tw_t \tag{2}
\end{equation}

The technology in the economy is a standard Cobb-Douglas with CRS:

\begin{equation}
Y = K^{\alpha}L^{1-\alpha}\tag{3}
\end{equation}

The FOC of the utility-maximization problem are:

\begin{align}
 c_{t+1} &= c_{t} [\beta (1+r_{t+1})] ^{\frac{1}{\sigma}}\tag{4}\\
 %
 c_{t}^{\sigma} l_{t}^{\gamma} &= w\tag{5}\\
 %
\end{align}

While the FOCs of the profit-maximizing problem are:

$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{6}$$

$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{7}$$

In the steady state, equation (4) implies the following

\begin{equation}
r   = \frac{1-\beta}{\beta}\tag{8}
\end{equation}

Hence, to solve for the steady state, we need to solve for the following system of equation:

\begin{equation}
C = rK + wL \tag{9}
\end{equation}

\begin{equation}
C^{\sigma} L^{\gamma} =w\tag{10}
\end{equation}

$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{11}$$

$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{12}$$

Where the unknowns are $C, K, L,w$ ($r$ is known).

But from here I can't solve for the exact values of these variables. What information I miss?

1) Instead of $(9)$ use

$$C = K^{\alpha}L^{1-\alpha} \tag{9*}$$

2) Equation $(10)$ is not correct. For constant population $N$ and identical agents,

$$c^{\sigma} l^{\gamma} =  w \implies (C/N)^{\sigma} (L/N)^{\gamma} =  w \implies C^{\sigma} L^{\gamma} = N^{\sigma +\gamma}w \tag{10*}$$

Insert $(9^*)$ in $(11)$ and solve for $C$ as a function of $K$, using $(8)$ for $r$. Substitute back in $(9^*)$ and solve for $K$ as a function of $L$. Insert into $(12)$ and you will get $w$ as a function of parameters. Etc.


Why can't the Land-Value Tax (in some cases) be passed on to tenants? According to https://en.wikipedia.org/wiki/Land_value_tax:

  Because the supply of land is essentially fixed, land rents depend on what tenants are prepared to pay,

That doesn't make any sense.  If more people want to live in a particular place, rents will increase.

  rather than on landlord expenses, preventing landlords from passing LVT to tenants.

Only if there's somewhere else that tenants want to be, that also has lower rents.  The real estate boom in San Francisco shows that when people with lots of money really want to live in a particular place, rents rise.

Thus, what stops the Land-Value Tax be passed on to tenants (in the situation where demand outstrips supply, and the government decides to increase the LVT)?

Because the supply of land is essentially fixed, land rents depend on what tenants are prepared to pay,

  That doesn't make any sense. If more people want to live in a particular place, rents will increase.

And if rents increase, then obviously tenants are prepared to pay more.  If they aren't willing to pay increased rents, then they don't really want to live there more than another place.  I.e. your statement is entirely consistent with the statement that you are criticizing.

    rather than on landlord expenses, preventing landlords from passing LVT to tenants.

  Only if there's somewhere else that tenants want to be, that also has lower rents. The real estate boom in San Francisco shows that when people with lots of money really want to live in a particular place, rents rise.

I think you missed the point here.  Presumably landlords are already charging what the market will bear.  How does a Land Value Tax (LVT) change what the market will bear?  As it's a tax on landlords, obviously it has no direct effect on tenants.  The tenants are not prepared to pay more rent as a result of the LVT.

If rents will go up by \$100 a month without an LVT and \$100 a month with an LVT, then landlords can't pass the LVT to tenants.  That's true even if the \$100 is enough to pay the LVT.  Without the LVT, the landlord would have raised the rent and kept all of the money (minus income taxes, etc.).  With the LVT, the landlord pays the LVT and may still have to pay more income taxes (depending on the deductibility of the LVT).

The only way that an LVT can influence rents from the landlord side is if it causes the number of units available for rent to either increase or decrease.  In the long term, that happens.  Landowners can choose to build more or fewer units on their land.  But in the immediate term, the number of units is fixed.  A landowner can't take a single family home and instantaneously convert it into a hundred-unit apartment building, even if there is sufficient land to support that.  Thus the LVT does not change unit availability or the rent that tenants are willing to pay.

Now, it is possible that the LVT enables spending that tenants find valuable.  And that spending causes demand and therefore rents to increase.  But that spending would be just as valuable to the tenants if it were financed by a different tax with less impact on landowners.  Of course, tenants might be negatively impacted by that tax, e.g. if they pay it.  But those effects are aside from the impact of an LVT on landlords.

The LVT itself does not increase rents in the short term.  It may increase rents in the long term if it drives away landowners by making some places economically infeasible to rent.  But that assumes that landlord profits are less than the LVT.  Or landowners may choose not to expand the number of units in the long term.  Or the LVT may finance a benefit that increases the value of the units to tenants.

So the landlord can only pass on the LVT if tenants benefit from the spending financed by the LVT, shifting the demand curve such that tenants will pay more.  So the more that the LVT finances benefits for tenants, the better able the landlord will be to pass on the LVT.

If those things don't happen, the landlord can't pass on the LVT.  The landlord can only get the regular market rent.  That may be an increasing market rent, but the increase is not due to the LVT.  It's only determined by movement in the tenants' demand curve, which is not directly impacted by the LVT.


Why'd the lender suffer the collateral haircut, not the borrower? In the quote below, why'd Steve offer $10.5, when your Air Jordans are worth \$11? Why wouldn't Steve (the lender) assign the collateral haircut to you, and require you to pay him \$11 + risk premium?

Apologies for the lengthy quote : please advise how I can shorten it.

ELI5: Repos, reverse repos, bonds & how they correlate. : investing

  It's lunch time and you're sitting in the cafeteria wearing your special edition purple and gold Air Jordans when you realize that you forgot your wallet at home. You need your wallet so that you can buy food. Or maybe you need your wallet so you can repay Jimmy for the sandwich he bought you last week.

  You got to your friend Steve who comes from a rich family and is always carrying extra cash around. You want to see if Steve will lend you some money if you promise to pay him back tomorrow. Steve is no dummy--he knows if you forgot your wallet today you might forget it tomorrow. So he offers to give you \$10 for lunch if you pay him back \$11 tomorrow. The extra \$1 is interest you pay Steve for trusting that you will pay him back.

  \$1 seems like a lot of money for one day to you. But you do need to eat so you don't have a lot of options. You don't have your wallet. But you do have your Air Jordans. Let's say your Air Jordan's are worth exactly \$11.

  You offer to let Steve hang on to your Air Jordans as a guarantee that you will pay him back tomorrow. If you don't pay him back--he gets to keep your shoes. The Air Jordans are what is called collateral. Steve agrees to this because he knows if you don't pay him back, someone else will buy the Air Jordans from him immediately for a market price of \$11.

  $\color{red}{\text{Steve offers you a new deal.}}$ In exchange for holding on to your Air Jordans, he will give you (immediately) a \$10 bill with the obligation that you will buy back the Air Jordans from him tomorrow for \$10.50.

  Two things have happened here. Firstly, Steve is giving you \$10 to hold on to an \$11 pair of shoes. How can he get away with this? Well--Steve is assuming the risk that Nike announces overnight that the Air Jordans are undergoing a recall and the value goes to \$0. Exchanging shoes as collateral for a loan at less than the market value is called a collateral haircut. The haircut is meant to account for the risk that the collateral asset might lose considerable valuable before the end of the loan. In this case the haircut was 1/11 = 9.1%.

  Next, you are still paying \$10.5 back to Steve tomorrow. This new rate of interest (\$.5/\$10 = 5%) is called the repo rate. This is 50% smaller than the loan Steve offered without a collateral backing (\$1/\$10 = 10%). Steve is still charging a "premium" for the possibility that you don't pay him back tomorrow (a default) and the opportunity cost he is experiencing by not investing his \$10 in something else.

  This entire transaction (exchanging the Air Jordans for \$10 and repaying \$10.50 the next day and receiving back the Air Jordans) is called an overnight Repo transaction. Overnight is indicative of the term of the repo. A term repo could be an agreement over a longer period of time.

  In this case, Steve has performed a reverse Repo. Typically dealers and other large financial institutions use repos to generate liquidity--getting cash (\$\$) in exchange for an asset (Air Jordans)-- or to achieve leverage by using their current assets to secure a loan that is used to buy another asset-- using the cash from the Air Jordans repo to buy a second pair of Air Jordans--.

Why'd the lender suffer the collateral haircut

This is not an accurate way of describing what is happening. No one "suffered". The collateral was entered at lower than current market value, to account for the risk. It can also be repossessed from Steve at lower than current market value.

  Why wouldn't Steve (the lender) assign the collateral haircut to you, and require you to pay him $11 + risk premium?

That would have also been acceptable to Steve. As it is explained in the text

  $1 seems like a lot of money for one day to you. But you do need to eat so you don't have a lot of options.

So it is in fact you who did not want to pay the risk premium, and Steve went along with your alternate offer.


'No arbitrage' vs 'Efficient market' I am trying to wrap my head around the 'no arbitrage' (NA) and 'efficient market' (EM) assumptions, in particular their difference.

Please correct me if I am wrong, but NA just claims that there is no trading strategy that generates profit but has zero risk, while by contrast EM states that the (historical) price data of an asset already contains all that there is to know about its future development.

Does one of them imply the other?

NA, in my opinion, is the lack of a trading strategy to generate excess returns, that means returns greater than those of assets in the same risk category. Note that it doesn't mean no risk.

A market is efficient, again in my own understanding, when all the contents of an information set are reflected in the market's assets prices. There are 3 forms of market efficiency weak, semi-strong and strong.

Weak form efficiency is given when the information set contains historical data about the assets, for example past prices, returns, past news about the companies, etc. An example of weak form efficiency would be when past growth data of a company is incorporated into its equity prices.
Semi-strong form efficiency occurs when the market is weak form efficient and the information set also includes all currently available public data about the assets in the market. For example press releases, macroeconomic indicators, etc. An example of semi-strong efficiency would be when due to an increase in the interest rate, the cost of capital of a company increases and the equity prices fall.
Strong form efficiency takes place when a market is semi-strong form efficient and the information set also includes all available private information about assets in the market. For example, plans yet not publicly available about a new product release or private events in the lives of the members of the management team. An example of strong form efficiency would be if, due to the CEO of the company going through a divorce, the company's shares prices fall.


How to find a firm's cost function based on its production function The question
A firm’s production function is given by
$$q=F(L,K)=L^{1\over{4}}K^{1\over{4}}$$
find the firm's cost function $C(w,r,q)$,
What I know so far
I'm aware that the Technical rate of subsitution (TRS) is:
$${MP_L\over{MP_K}}={{\partial{F}\over{\partial{L}}}\over{\partial{F}\over{\partial{K}}}}={K\over{L}}$$
but I'm not sure how this can help me find the cost function

A firm's production function is a relationship between inputs and outputs (thus what are the combinations of input and out that can be obtained given the available technology). y-axis is Output and x-axis is for inputs.
A firm's cost function is a relationship between costs (expressed in monetary terms) and output. Thus, for every level of output, and market prices for inputs, what is the cheapest way to produce it. Output on the x-axis and Cost (monetary units) on the y-axis.

This relationship isn't as straight forward if prices aren't linear. However, most models will assume linear prices. If that is the case, you can kind of see the cost function as the inverse of the production function (essentially flipping the axes), because more inputs for an unit of output imply more costs.

As far as actually getting the cost function from the production function, you have to start by finding the argmin for this cost minimization problem:
$$\min (w \cdot k + r \cdot k)$$
s.t:
$$
q=L^{1/4}K^{1/4}
$$
F.O.C.

$w={\partial{F}\over{\partial{L}}}=f_{L}(L,K)$

$r={\partial{F}\over{\partial{K}}}=f_{K}(L,K)$

So wages will be equal to marginal productivity of Labor and interest rate will be equal to marginal productivity of capital in equilibrium.
Let me know if you don't know how to keep going from here.


What's the status on uncertainty shocks? Bloom (2009) analyzes the impact of uncertainty shocks, and Bloom et al (2014) propose a business cycle model based on these shocks.

It seems to be a young field, but is there any consensus on strength and weaknesses of this new shock? Is it the answer to some puzzles such as the risk premium puzzle?

What is the opinion of the literature about this field, and is there already a handbook chapter on this?

Macroeconomic Uncertainty and Expected Stock Returns is a recent paper (Bali, Brown, and Tang (2015)) finds an economically important effect on uncertainty on returns.

  This paper introduces a broad index of macroeconomic uncertainty based
  on ex-ante measures of cross-sectional dispersion in economic
  forecasts by the Survey of Professional Forecasters. We estimate
  individual stock exposure to a newly proposed measure of economic
  uncertainty index and find that the resulting uncertainty beta
  predicts a significant proportion of the cross-sectional dispersion in
  stock returns. After controlling for a large set of stock
  characteristics and risk factors, we find the predicted negative
  relation between uncertainty beta and future stock returns remains
  economically and statistically significant. The significantly negative
  uncertainty premium is robust to alternative measures of uncertainty
  index and distinct from the negative market volatility risk premium
  identified by earlier studies.

Dynamic co-movements of stock market returns, implied volatility and policy uncertainty (Antonakakisa, Chatziantonioua, and Filisb (2013)) finds that policy uncertainty, as measured by Baker, Bloom, and Davis (2013), has an unconditional correlation with the Vix of about 0.43. But using a DCC_Garch they find episodes when this correlation was negative, which may be surprising.


Does every structural policy change count as structural break? I was reading the answers to this question when, the one by Dawson Verley aroused my curiosity. Does every change in structural policy result in a structural break? What conditions are necessary and/or sufficient for break?

Any help would be appreciated.

Assuming I understand the question appropriately, the answer, generally, is yes - but in its application to times series analysis/forecasting not all "structural breaks" would end up being included in a given model.  For example, if we are modeling labor supply, policy related to minimum wage would be hugely important (an increase can be seen as a structural break or a shock).  However, a small tariff placed on sugar imports, though it may effect labor supply in some minute way in some region, would not constitute a structural break in consideration of our model (though, by definition, it is a break).

To answer the second part of your question related to what conditions would necessitate inclusion in a model, I would argue that it is an intersection of theory/logic and statistical relevance.  If the influence of the break on other variables is virtually unobserved, or if its coefficients are far from significant then it need not be included (barring some other compelling case for its inclusion).  Or, we could say during the modeling process some break/shock variable added no advantage to the model fit (thinking AIC or SIC/BIC here).

Hopefully that at least gets at your question.


Limited Liability in Agency Problem Consider a simple one-period model where a creditor provides $I$ as investment to a firm and firm produces $\pi_G$ or $\pi_B$ depending on its performance. Let $R_i$ the portion of the profit that the firm transfers to investor, where $i=\{G,B\}$.

My Questions

(1) If we enforce limited liability structure in an agency problem, why is it that:

$$\pi_i\geq R_i.$$

(2) In an agency problem, is it reasonable to have this assumption? Why have limited liability as constraint at all?

(1) The limited liability constraint ensures that the agent does not pay out of her own pocket the investors dividends. The investor can only take as much as the company made, and cannot charge the agent money she did not produce.

(2) It is reasonable in a sense. In the real world, we believe investors cannot force a manager to pay for bad performance from her own bank account (at least I believe it).

Relatedly, on a technical note, limited liability is necessary to make problems "interesting" in Principal-Agent models. It is well known that is both agent and principal are risk-neutral, the principal "sells the firm" to the agent, and the agent does the optimal thing, and we actually do not have a moral hazard problem, in the sense that we can achieve first best.

To make the problem interesting, we usually assume either:

(i) the agent is risk averse or

(ii) we have limited liability.


Annual Data and Heteroscedasticity (Engle's ARCH test) GARCH models are often applied to financial time series (daily, weekly or monthly stock returns). What about lower frequency such as quarterly and annual time series? This could include macroeconomic variables (e.g. GDP, aggregate consumption) or accounting data (e.g. book values, ...)
Question: Is there evidence for time varying second moments in annual economic data? Are there papers applying ARCH tests at low frequencies? What lags are commonly used in this case?
I'm mostly curious whether/how working at a much lower frequency impacts the ''standard'' GARCH modeling from daily stock returns?

Is there evidence for time varying second moments in annual economic data?

Yes, although not that much in finance in particular but in economics in general resounding yes. For example, the highly cited Engle (2001), GARCH 101: The use of ARCH/GARCH models in applied econometrics, besides examples with daily data refers also to some examples with quarterly data. Or more directly an example of this that actually uses low frequency data (yearly/quarterly/monthly) and uses variant of GARCH, GARCH-SK,  is Narayan and Liu (2018). Or another direct example is Kontonikas (2004) who applies GARCH-M models to quarterly and monthly inflation data. So GARCH is and can be to many economic series on quarterly or annual frequency.
Usually people go for quarterly data or yearly data when it comes to macro variables like GDP, unemployment, inflation etc and for higher frequencies when it comes to finance (e.g. stock prices, bitcoin etc). This is because typically having high frequency means more data. However, at the same time having more data is not always better. For example, having access to GDP data on daily frequency would likely just add noise so there is no point in trying to go for high frequency data there. However, the same does not necessarily hold for variables like exchange rates or stock prices. What the optimal frequency for particular research is has to be to an extent decided on case by case basis. Also GARCH models are somewhat more common with high frequency data as they are primarily used to solve issues like volatility clustering, which will be more common problem at higher frequencies, but as the references above show application to lower frequency data is not unheard of.

What lags are commonly used in this case?

Lags in GARCH models do not depend on frequency of data per se (of course it is more likely there will be more autocorrelation in high frequency data than in low frequency data). You include lags in order to appropriately model autocorrelation. This will be case dependent whether you have high frequency data or low frequency data. This being said evidence shows that often it is hard to beat GARCH(1,1) (see Hansen et al. 2001) and this often holds for various frequencies*, but if you would want to do it 'by the books' you should just test for this and in addition you can potentially estimate several models with different $p$ (and $q$ as well even if that was not part of your question) and compare their performance (it is really not that much extra work once you write code for the main model).

* Although note the paper has received some criticism and so you should not just necessarily take it at face value - see here


Interpretation of ACF and PACF First, I am a French student, so forgive me for my English which can be not clear at all.

I have to analyze a financial series. I have some difficulties to make the second part of the work which focuses on ARMA model. I can't read (interpret?) my Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) correlogramm.

To my mind, here we have a MA(0) but I don't know if it's possible. In this case, what does it mean ?

And I don't know also with the PACF graph, if we are in a AR(3) or AR(6). And why the first stick is not over the "confidence level" ?

It is possible that my interpretations are false, please tell me.

In some statistical software (not all), in a correlogram the "zero lag" is also depicted - but the zero lag is just the correlation of a random variable with its own self, so it is by construction equal to unity. This appears to be happening in your first graph where the autocorrelation function is calculated.

The second graph is the partial autocorrelation function which calculates the correlation coefficients after the effect of all "previous" lags (i.e. of lower order) has been removed (by linear projection estimation).

First impression? The process is white noise. What lags appear to exceed the  "statistical significance threshold", they nevertheless indicate very small values, never greater  than $0.05$. This is truly economically negligible correlation even if it exists (irrespective of whether it is "statistically significant"), and most likely is due to "finite-sample variation", rather than indication of any actual relation.

And apart from that, do the lags appearing "significant" make sense?
It always matters what the process you are examining is, and what is its "typical properties" based on past scientific work (but don't take this to mean that "past typical properties" always carry over to all the realizations of the process).

In any case, it would be instructive to "follow blindly" the Correlograms and see what you get, by trying various combinations based on the lags that appear statistically significant.


Mixed Strategies in Bayesian Games I am confused about the following statement from Wikipedia:

"A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player."

How do we check the Nash equilibrium inequality then?

For example, in Bayesian games with player set $N=\{1,2,\ldots,n\}$, the equilibrium is defined as the strategy profile $s=(s_i,s_{-i})$ such that, we have $$EU_i(s_i,s_{-i})\geq EU_i(s_i',s_{-i}),$$ for all $s_i'\in S_i$, for all $i\in N$ and $s_i'\neq s_i$.

In many texts, $S_i$ is defined as the function $S_i:\Theta_i\to\Pi(A_i)$, where $\Theta_i$ is the type set and $\Pi(A_i)$ is the probability distribution over the actions.

How is it possible to have infinitely many mixed strategies? What is $S_i$ exactly, a set or a function?

Since there are infinitely many numbers in $[0,1]$, agent $i$ has infinitely many probabilities to randomize over actions in their action space $A_i$. So far this has nothing to do with the game being Baysian.  In a Baysian game, a strategy $\sigma_i: \Theta_i \rightarrow \Pi(A_i)$ (or $s_i$ instead of $\sigma_i$) maps a type into a probability distribution over (pure) actions. You can call the space of all such functions $\Sigma_i$ (or $S_i$). There are many other notational conventions. So you always have to take care of careful definitions.
Just because there are infinitely many ways to randomize over the actions (pure strategies) does not mean that you have to compare all of them one-by-one in an inequality like yours. You can invoke monotonicity arguments or simply maximize over the probabilities.
In a similar vein, you can find an extremum of the function $x^2$ for $x\in[-1,1]$ although there are infinitely many numbers $x\in[-1,1]$.


Why life expectance is growing faster than retirement age? - source needed Need sources of information about "Why is life expectancy growing faster than retirement age?"

Is there any literature about "why the increased quality of life and medicine in rich countries doesn’t increase the ability to work for older people?"

Loss-aversion and political tactics.

A lot of the elderly are no longer physical broke down - which was the original reason for retirement. But, if you can retire in good health, it is completely rational for the individual to do this, and to vote for the politicians who will provide this for you.

Or, as a danish mayor once put it:

  "When you first throw something into the monky cage, it is hard to
  get it back again"

Aka, when you first give people major entitlements and life improvements, it is next to political impossible to take them away.


Gross substitutes vs. net substitutes Wikipedia explains the difference between products that are "gross substitutes" and products that are "net substitutes". However, the mathematical explanation doesn't give much intuition about these concepts. So my questions are:

What is the intuitive difference between gross and net substitutes?
What are good real-life examples of these?
Why is the requirement for competitive equilibrium, "gross substitute" and not "net substitute"? I.e., why does a competitive equilibrium not exist if the products are net-substitute?

Intuitively, a higher price for pears means that I have to give up more apples to be able to afford an extra pear (or, conversely, if I give up one pear then the number of extra apples that I can afford increases). This is going to make me want to reduce my pear consumption and increase my apple consumption (in orther words, to substitute away from pears towards apples). Graphically, this corresponds to a change in the slope of the budget constraint and is known as the (Hicksian) substitution effect.

This substitution effect is, however, moderated by a second consideration. If I increase one or more prices then the total amount of "stuff" that I can afford to buy decreases, so it is as if my income has decreased (which would correspond to a shift of the budget constraint towards the origin). This is known as the income effect and will usually mean that my consumption of apples decreases if the price of pears rises.

When the income and substitutes effects are put together, you get the total effect of an increase in the pear price upon the demand for apples. This total effect gives rise the the notion of gross substitutes: apples and pears are gross substitutes if the following is true

  increasing the price of pears causes the consumer to demand more apples.

Formally, this is written as $$\frac{\partial X_a}{\partial p_P}>0,$$ where $X_A$ and $p_P$ are the demand for apples and the price of pears respectively.

To arrive at the notion of net substitutes we simply take a price change and shut-down the income effect. The hypothetical exercise works like this:

we increase the price of pears, which induces both an income effect and a substitution effect.
to compensate the consumer for the income effect, we give him exactly enough extra money to ensure that (even though one of the prices is higher) he can still afford to get onto the same indifference curve as before. Hence his utility will remain unchanged.
We look at how his demand changed, which now depends only on the substitution effect (since we have compensated the consumer for any change in real income).

Two goods are net substitutes if, after making this adjustment, we find that the demand for apples has increased. Thus, we say that two produces are net substitutes if

  Increasing the price of pears while compensating the consumer for the resulting decrease in his real income causes the consumer to demand more apples.

Formally, $$\left.\frac{\partial X_a}{\partial p_P}\right|_{\text{constant }U}>0,$$ where $X_A$ and $p_P$ are the demand for apples and the price of pears respectively, and $U$ is the consumer's utility.

It is not true to say that "a competitive equilibrium [does] not exist if the products are net-substitute". Indeed, most of the time, products that are gross substitutes will also be net substitutes as well. Thus, most examples of gross substitute preferences supporting a competitive equilibrium will also be examples of net substitutes doing the same.

The reason why it doesn't make sense to state the existence condition for competitive equilibrium in terms of net substitutes is that net substitutes is a purely hypothetical construction in which a fictitious agent intervenes to shut down the income effect and keep the consumer's utility constant. The whole point of a competitive equilibrium is that there is no such intervention: the equilibrium is entirely decentralised and is sustained purely by finding prices such that the market clears when consumers pick their optimal demand.


Does Visitor Export Include Airline Tickets? Does visitor export include the purchase of airline tickets? If so, I would assume that it only counts airline businesses that are established in the country of question.

The UN SNA (2008, p. 42) has this to say:

  planes belonging to a domestic airline are always assets of the domestic economy regardless of where in the world they happen to be.

So, conceptually,* we should include in our country's GDP any services provided by any planes belonging to domestic airlines. And so for example, the fare paid in a Ryanair (based in Ireland) one-way ticket from Paris (France) to Berlin (Germany) would be counted in Ireland's GDP.

Conceptually, an export is any good or service sold by residents to non-residents. So, if the above one-way ticket were purchased by an Irish resident, then it should be counted as consumption in Ireland's GDP. If it were instead purchased by an Irish non-resident, then it should instead be counted as exports in Ireland's GDP. (I don't know though if in practice they actually make this distinction.)

*In practice, I am not sure how exactly this is done by national statistical agencies. My suspicion is that they probably don't consider each ticket sold and instead just look at aggregate figures provided by domestic airlines, e.g. their profits.


Indifference Curve Analysis I would like to analyse how COVID-19 has impacted the aviation industry by looking at how the demand for airlines + holidays has fallen via an indifference curve analysis. However, I'm not sure where to start
Would the indifference curve shift inwards?

Indifference curves represent preferences. Preferences are usually assumed to be stable, i.e. they do not change. So no, indifference curves don't shift.


Flow and Time derivative? If $a_{t}$ is the asset at time $t$, $c_{t}$ the consumption at time $t$, and $r$ the interest rate, then $a_{t+1} - a_{t} = ra_{t} - c_{t}$ is the flow of $a$. But it seems that in macroeconomics people just write $\dot{a}_{t} = ra_{t}-c_{t}$ instead in continuous time setting. I do not see the passage from $a_{t+1}-a_{t}$ to $\dot{a}_{t}$ in general. I can justify it in some special cases. For instance, if $a_{t+1}-a_{t} = r$ for some $r$, all $t$, then $a_{t+h}-a_{t} = hr$, so taking limit of the difference quotient leads us to $\dot{a}_{t} = r$. But this depends on the fact that the right-hand-side term is linear in $t$.

I am unsure how economists deal with the transition from continuous time to discrete time. My background is in systems engineering/applied mathematics, and the economics literature I am familiar with is either discrete time or continuous time. Unfortunately, I have no good references on the conversions from discrete to continuous time from the systems engineering literature (got rid of the textbooks).

The key in this case is that the “a” coefficient in these systems do not necessarily correspond to the same number. We need to adjust it so that behaviour lines up.

If we are transitioning from continuous time to discrete time, we effectively are sampling the continuous time system. The frequency we sample at matters.

This is easy to see. Imagine that a continuous variable grows by 1% in six months. If the discrete time step is six months, the variable grows by 1.01 each period. However, if we our sample is annual, we get a transition growth step of $(1.01)^2$ per period. Essentially, the formula to convert the growth coefficient is similar to changing interest rate quote conventions from continuous compounding to a bond yield equivalent.

Although a continuously growing variable (like a compunding deposit) is relatively straightfoward to deal with, something like consumption series is more difficult to specify. If we want the discrete time to correspond to the sampled values of the continuous time version, we need to pin down the continuous time dynamics in some fashion. Is it constant during the time interval, or is it also growing in an exponential fashion (which would be the case if it were an endogenous variable inside a linear model)?

Going from discrete time to continuous time is more awkward. A continuous time system contains much more information than a discrete time system. There is an infinite number of continuous time systems that can generate the same discrete time system. We need a machanism to pin down the dynamics of the continuous time version, but the decision can be viewed as arbitrary. This is a topic that is not heavily studied in systems engineering, as physical systems are already truly operating in continuous time.

A simple economic example of the loss of information is as follows. Imagine that we have a quarterly model frequency, but a household receive a cash flow at some point during the quarter. Interest compounds daily (like a bank deposit). The final amount held at the end of the quarter depends upon the length of the holding period, which could be from 0-3 months in length. Since the discrete time model has no notion of times in between the quarterly sample periods, it has no way of modelling this timing effect that is a property of the continuous time model. This means that a discrete time model is likely only going to approximate a continuous time model.


Per Capita Income data of states in the USA Where can I get Per Capita Income data of states in the USA? Also which are the sources often used in econometric or macroeconomic analysis and how reliable are they?

BEA Person Income per capita for USA or states
https://apps.bea.gov/iTable/iTable.cfm?acrdn=5&isuri=1&reqid=70&step=1
St Louis FED can pull up this data for you.
https://fred.stlouisfed.org/series/A792RC0A052NBEA

For your second question, it really depends on how you define personal income.  A recommendation is to check the statistical agencies definition of personal income and see if it matches up with your analysis/ project.


Bayesian Nash Equilibrium in a Duopoly Cournot Competition I am having a hard time to solve a Bayesian Nash equilibrium game in a duopoly cournot competition setting. So, I have two firms with given production quantities, let's say $q_1$ and $q_2$ (i.e., not a decision). Firm 1's product is low quality whereas Firm 2's product is high quality with probability $\theta$ and low quality with $1-\theta$. As a result, different quality products have different market potential for Firm 2. The way I set up the price function for low quality is
$$M-q_1-q_2$$
For high quality products, there is a price premium such that the price for high quality is given by $$\alpha M-q_1-q_2$$ where $\alpha>1$. Another different aspect of the problem is that companies first sell products in the primary market under the price mechanism specified above. There exist a secondary market. It works in a way such that if any firm sells his (remaining) products there, he gets a guaranteed price $m$ ($M > m$).
Currently my solution involves 8 different cases. I solve Bayesian Nash equilibrium by solving three profit functions simultaneously to find the quantities to supply in the primary market. Since $q_1$ and $q_2$ are treated fixed, I need to have a $\min \{q_i, \hat{q}_i\}$ for $i=1, 2H, 2L$, where $\hat{q}_i$ is the equilibrium solution for the primary market. But I am trying to narrow the cases down to a small set so that I can continue my other analysis with the solution. Accordingly, I have two questions: 1. Is my current solution correct? 2. How can I narrow cases down to a manageable number?
If you wonder what my equilibrium solution looks like for primary market, here it is.
$$\hat{q}_1 =\frac{M-m}{3}-\frac{\theta (\alpha M-M)}{3}$$
$$\hat{q}_{2L} =\frac{M-m}{3}+\frac{\theta (\alpha M-M)}{6}$$
$$\hat{q}_{2H}^M = \frac{\alpha M-m}{3}+\frac{(1+\theta)(\alpha M-M)}{6}$$

If the capacities $q_1$ and $q_2$ are the same, say $\overline{q}$ you can rule out some cases.
The best response functions are given by:
$$
\begin{align*}
\hat q_1 &= \min\left\{\frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2}, \overline{q}\right\},\\
\hat q_2^L &= \min\left\{\frac{M - \hat q_1 - m}{2}, \overline{q}\right\},\\
\hat q_2^H &= \min\left\{\frac{\alpha M - \hat q_1 - m}{2}, \overline{q}\right\}
\end{align*}
$$
First we can show that $\hat q_2^H \ge \hat q_2^L$.
Proposition: $\hat q_2^H \ge \hat q_2^L$

Proof: If $\hat q_2^H = \overline{q}$ the proof is obvious. Else, from $\alpha > 0$, we have:
$$
\hat q_2^H = \frac{\alpha M - \hat q_1 - m}{2} \ge \frac{M - \hat q_1 - m}{2} \ge q_2^L.
$$

Next, we can also show that $\hat q_2^L \ge \hat q_1$.
Proposition: $\hat q_2^L \ge \hat q_1$.

Proof: if $\hat q_2^L = \overline{q}$ the proof is obvious. If not, then:
$$
\begin{align*}
\hat q_1 - \hat q_2^L &\le \frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2} - \hat q_2^L,\\
&= \frac{M - (1-\theta) \hat q_2^L - \theta \hat q_2^H - m}{2} - \frac{M - \hat q_1 - m}{2},\\
&= \frac{-(1-\theta) \hat q_2^L - \theta \hat q_2^H + \hat q_1}{2},\\
\to 2 \hat q_1 - 2 \hat q_2^L &\le -(1- \theta) \hat q_2^L - \theta \hat q_2^H + \hat q_1 \\
\to \hat q_1 &\le \hat q_2^L - \theta(\hat q_2^H - \hat q_2^L) \le \hat q_2^L.
\end{align*}
$$
where the last line follows from $\hat q_2^H \ge \hat q_2^L$.

The two propositions show that we only have to consider the following cases:

$\hat q_2^H < \overline{q}$ and $\hat q_2^L < \overline{q}$ and $\hat q_1 < \overline{q}$
$\hat q_2^H = \overline{q}$ and $\hat q_2^L < \overline{q}$ and $\hat q_1 < \overline{q}$.
$\hat q_2^H = \hat q_2^L = \overline{q}$ and $\hat q_1 < \overline{q}$.
$\hat q_2^H = \hat q_2^L = \hat q_1 = \overline{q}$.

For each of them you can compute the equilibrium values and then you need to check if they indeed correspond to the best responses.
This reduces the number of cases from 8 to 4. I don't think you can do better.


Can one bank be the correspondent bank of itself? http://www.investopedia.com/terms/c/correspondent-bank.asp

Correspondent bank is like a delegator in a country to operate on behalf of the delegatee who doesn't normally has established business in that country.

However, I heard that for some bank groups, one subsidiary can be the correspondent bank for another subsidiary, especially in the area of clearing. Let's say Deutsche Bank Italy can be the correspondent bank for Deutsche Bank North America.

Does it imply that Deutsche Bank Italy can hold a Nostro account in Deutsche Bank North America or vice versa.

I wonder why would these kind of relationship considered as correspondent? What are the limitations/benefits for banks operating in this way?

Yes it is possible, I worked for a big bank subsidiary in a different country. We had a nostro account with the parent bank, and with its other subsidiaries. I cannot comment on the benefits/limitations of this though.

EDIT:
I remember we did not pay fee for the nostro accounts held with the parent and subsidiary banks. Moreover, these accounts were necessary as we had to deal with transactions involving various currencies. If let's say a client wanted to deposit in EUR, the bank should have a EUR nostro otherwise, the bank would not accept euro deposits.


Why min AC = min SRAC at the minima of AC curve? At the lowest point of long run average cost curve AC, the SAC is also at its minimum and that is not the case with any other SAC curve bounded by the envelope. Why is that?

Short-run implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$.
Given a value $q$ let us denote the cost minimizing value of $x$ by $x^*$, i.e.
$$
x^* = \arg\min_x SC(x,q).
$$
Cost minimization implies that in the long-run this will be the chosen value of $x$, thus
$$
C(q) = SC(x^*,q) = \min_x SC(x,q).
$$
Dividing by $q$ we get
$$
AC(q) = SAC(x^*,q) = \min_x SAC(x,q).
$$
The above holds for any $q$.


Finding Optimal Bundle and Change in Satisfaction after Changes in Budget Constraint I am looking at the following exercise and struggling with the solution proposed by my microeconomics book.

  A consumer spends all his income on two goods, X and Y. The prices he paid and the quantities he consumed last year are as follows: PX = 15, X = 20, PY = 25, and Y = 30.
  We assume the consumer's preferences follow the 5 properties for preference ordering (complete, transitive, more-is-better, convex, continuous).

  If the prices next year are PX = 6 and PY = 30, and the consumer's income is 1,020, will he be better or worse off than he was in the previous year? (Assume that his tastes do not change.)

For the Solution we first find the consumer's budget constraint of last year:

M = PX * X + PY * Y = 15 * 20 + 25 * 30 = 1050

We need to ask can he buy last year’s bundle given the new prices?

PXnew * X + PYnew * Y = 6 * 20 + 30 * 30 = 1020

Given that the consumer can still afford last year's bundle, he must be (in my opinion) at least as well off. I am working with a economics book and the solution proposed in the book claims, that the consumer must be better off using the following figure as explanation:

I am wondering, couldn't the indifference curve that is tangent to the bundle at X = 20, Y = 30 for the last year, also be tangent to a point on the new budget constraint and equal the highest indifference curve that can be reached?

Is there an explanation, given the 5 properties of preference ordering that the consumer MUST be better off.

EDIT:

I have added the image below to clarify my question.

I am wondering whether the indifference curve in red in the image could look like this and imply that the satisfaction level is not increasing as the consumer will stay on the same indifference curve but consume a different combination of X and Y.

I understand what you though, the old indifference curve sure can touch but cannot conclude to be the optimal one.

As the evidence of the money he can pay 1050 of the last year budget but for this year he could only use just 1020 to buy the same combination of those goods with the same utility so that means he still can buy and consume more with the relative purchasing power.

I will just use 2 assumptions that are "more is better", "no bliss point". It is said that the budget which has shown or given here is not significant to the level that the consumer will keep and not spend so that as he can spend he will and with "no bliss point" that means this consumption is too small and yet not to the level that he will have negative marginal utility.

Therefore, as he can buy the old combination with the lesser price so he can allocate the leftover budget for either of two goods.


Measure of expected inflation How is expected inflation measured for policy purposes? Is it based on some economic model or some survey?

There are three big sources for data on inflation expectations in the US:

Market implied expectations from the TIPS spread. TIPS securities provide inflation protection but are otherwise roughly comparable to normal treasuries, so the difference in yields between these two is a measure of expected inflation.
Consumer expectations from the University of Michigan Survey of Consumers, which you can find here.
Surveys of professional forecasters, such as this survey.

Which of these measures is most useful is not clear, and I've seen all three used at different times and in different places. This speech by Janet Yellen has some discussion of the different measures, as does this paper by Carola Binder.


Contract curve for firms with linear utility functions I am attempting to find a contract curve for a production economy with two linear utility functions.

Normally, I would find the point where the Marginal Rate of Technical Substitution were equal for the two firms. However, in this case, this does not seem to work (I end up getting an equation of two numbers).

How must I proceed?

Is it all places where both functions are equal?

1.) An economy with two agents who both have linear preferences will always have Pareto optimal points along either the right and bottom edges or the left and top edges of the Edgeworth box. Edges are dictated by the ratio of MRSa,MRSb.

2.) The contract curve is the locus of Pareto Optimal points.


Say's law stated in terms of general equilibrium theory? Say's law is roughly that "an increase in aggregate supply generates an equal increase in aggregate demand".

Can this law be stated, or derived from, a model of general equilibrium?

EDIT: what if we assume a dynamic economy? I.e. where saving is possible.

How about Walras's law?

  Walras's law is a principle in general equilibrium theory asserting that budget constraints imply that the values of excess demand (or, conversely, excess market supplies) must sum to zero.


Why traditional banks tend not to provide finance to the poor? As in title, quite a simple question - however looking for some more arguments and points of view. Mainly talking about the poor in developing countries, however if some arguments are more "applicable" to the poorer part of the society in already developed countries - would love to read them as well.

This is actually heavily studied question in development economics. Here are some common answers:

Lack of secure property rights in developing countries. More substantial loans require collateral, in many developing countries especially in rural areas people don’t have deeds of ownership for their houses or farms. Even when they have access to some deed the property rights might be not secure.
Lack of identification - this is related to the previous answer. In many developing countries it’s hard to prove that you are who you are. Again especially in rural areas people might not even have birth certificates. Hence, a bank who does not have local knowledge like loan shark might not get repaid because person just disappears. Hence the costs of private monitoring for classic bank are prohibitively expensive.
In some places it might still be physically very hard to reach the people or you can put it other way around people in many places can’t physically access banking services - due to lack of transport or internet.

Also I got these from the book Poor Economics by Banerjee and Duflo (recent Nobel prize winners) they go in their book into much greater detail of this issue, so I would recommend it and the papers cited therein for more info. Although the book deals with multiple issues so it’s not only about this - just mentioning it so you know what to expect if you plan to get it.


How to calculate GDP increase using Cobb-Douglas production function? I use this economics stack exchange for the first time. I have a question regarding the Cobb-Douglas production function. The question is this: GDP in an economy is growing at 3% a year in real terms. Population is constant. The government decides to allow a significant increase in immigration so that the population (and the workforce) starts to grow by 1% a year. Output is produced in the economy according to a Cobb-Douglas production function. The share of labour income in GDP is 70%. How much higher will GDP be as a result of the new immigration policy after 20 years? How much higher will GDP per capita be?

I am not be able to understand how to use the function in this problem. Sorry if I put a question in a wrong way. Thank you.

I think this approach might be useful.The Cobb douglas production function with constant returns to scale is
\begin{equation}
Q=K^{\alpha}L^{1-\alpha}
\end{equation}
For simplicity, we use logarithms for the rates of change of its determinants
\begin{equation}
ln(Q)=\alpha ln(K)+(1-\alpha)ln(L)
\end{equation}

Differentiating with respect to time on both sides of the equation

\begin{equation}
\frac{1}{Q}\frac{dQ}{dt}=\alpha\frac{1}{K}\frac{dK}{dt}+(1-\alpha)\frac{1}{L}\frac{dL}{dt}
\end{equation}

\begin{equation}
\begin{array}{l}
 \text{Growth rate of GDP:}\frac{1}{Q}\frac{dQ}{dt}=G_{Q} \\
\text{Growth rate of Capital:}\frac{1}{K}\frac{dK}{dt}=G_{K} \\
\text{Growth rate of Labor:}\frac{1}{L}\frac{dL}{dt}=G_{L}
\end{array}
\end{equation}
\begin{equation}
G_{Q}=\alpha G_{K}+(1-\alpha)G_{L}
\end{equation}
Where $\alpha$ is the capital share in the gross domestic product. In your case $(1-\alpha)=70\%$, hence $\alpha=30\%$

Thus,
$G_{k}=0\%$ (The capital in this case does not change.), $G_{L}=1\%$
Substituting in the original equation
\begin{equation}
G_{Q}=0.7\cdot 1\%=0.7\%
\end{equation}
for further growth rate.

Adding to the previous growth rate ($3\%$), we obtain
$3\%+0.7\%=3.7\%$
To find the new GDP after 20 years.
\begin{equation}
 (1+0.037)^{20}=2.068117
\end{equation}


Data on corruption in Venezuela I'm trying to find some data on the level of corruption in Venezuela, since Chavez came to power, and its evolution along time. I've searched the world bank, but they don't seem to have this data on Venezuela...

The Corruption Perceptions Index of Transparency International could work for you. Chavez was in power from 14 April 2002 – 5 March 2013 and the index is available from 1995-2015 and includes Venezuela. In addition to the cross-country data, there is a more detailed Venezuela page produced by Transparency International.


Shadow Price vs Marginal Cost Is there a difference between the two? If so can you intuitively explain it? I see internet resources saying these are equivalent, but I thought I remember someone telling me once shadow price was different than marginal cost, but I may be misremembering it. Bonus points if you can place the definition in the context of a non-equilibrium setting.

Marginal cost is always the same as the shadow price in the cost minimization problem
\begin{eqnarray*}
\min_x && w \cdot x \\
s.t. && f(x) = y.
\end{eqnarray*}
In optimum the shadow price (Lagrange multiplier) belonging to the condition $f(x) = y$ is the marginal cost.

However there are other optimization problems where the shadow price is something else, for example in the utility maximization problem

\begin{eqnarray*}
\max_{x,y} && U(x,y) \\
s.t. && p_x \cdot x + p_y \cdot y = m
\end{eqnarray*}

the shadow prices correspond to the marginal utility of money. (One could argue that this is the reciprocal of the marginal cost of producing more utility.)

Neither of these examples had anything to do with equilibrium, just individual optima.


Convex games: equivalence of definitions Let $N \subset \mathbb N$ denote the set of players and $v : 2^N \to \mathbb R$ , $v(\emptyset) = 0$, the characteristic function.
We call $(N,v)$ a cooperative transferable utility (TU) game.

Definition 1.
A game $(N,v)$ is called convex, if for all $S,T \subseteq N$ it holds that
\begin{align}
v(S \cup T) + v(S \cap T) \geq v(S) + v(T) \tag{C.1}.
\end{align}
In Hafalir (2007, p. 255) it says that Def. (1) is equivalent to the following statement.

Definition 2.
A game $(N,v)$ is called convex, if for all $S,T \subseteq N$ with $|S \setminus T| = |T \setminus S| \leq 1$ it holds that
\begin{align}
v(S \cup T) + v(S \cap T) \geq v(S) + v(T) \tag{C.2}.
\end{align}

There is no proof of the statement, but the author refers to Moulin (1988, p. 112). But here it only gives an equivalent for Definition 1.

Definition 1'.
A game $(N,v)$ is called convex, if for all $i \in N$ and $S \subset T \subset N \setminus \{i\}$ it holds that
\begin{align}
v(S \cup \{i\}) - v(S) \leq v(T \cup \{i\}) - v(T) \tag{C.3}.
\end{align}

Definitions 1. and 1'. are standard. I never heard of 2. though. I have a problem, however, where I can show that Def. 2. is satisfied. And now I'm wondering how to show the equivalence of Def. 2. with either 1. or 1.'.

Define the marginal contribution of $i \in N$ to any $C \subseteq N \setminus \{i\}$ by
\begin{align}
m_i(C) = v(C \cup \{i\}) - v(C).
\end{align}

We are going to show C.1 $\Rightarrow$ C.2 $\Rightarrow$ C.3 $\Rightarrow$ C.1.
Note that C.1 $\Rightarrow$ C.2 is trivially true.
If C.1 holds for all $S,T \subseteq N$, then C.2 must hold for all $S,T \subseteq N$ with the restriction $|S \setminus T| = |T \setminus S| = 1$.
In order to show C.2 $\Rightarrow$ C.3 consider any $S \subseteq T \subseteq N \setminus \{i\}$.
Let $T \setminus S = \{j_1, \ldots, j_k\}$ and define $S_\ell := T_{\ell-1} \cup \{i\}$ and $T_\ell := T_{\ell-1} \cup \{j_\ell\}$ for $\ell \in \{1,\ldots,k\}$ with $T_0 = S$.
Note that
\begin{align}
\begin{split}
T_{k-1} &~= T_{k-2} \cup \{j_{k-1}\} \\
&~= T_{k-3} \cup \{j_{k-2}\} \cup \{j_{k-1}\} = T_{k-3} \cup \{j_{k-2},j_{k-1}\}\\
& ~ \vdots\\
&~ =  T_0 \cup \{j_1,\ldots,j_{k-1}\}\\
&~ = S \cup \{j_1,\ldots,j_{k-1}\}\\
&~ = T \setminus \{j_k\}
\end{split}
\end{align}
Since $|S_\ell \setminus T_\ell| = |T_\ell \setminus S_\ell| = 1$ we can apply C.2, i.e.
\begin{align}
\begin{split}
&v(S_\ell \cup T_\ell) + v(S_\ell \cap T_\ell) \geq v(S_\ell) + v(T_\ell)\\
\Longleftrightarrow \quad &v(T_{\ell-1} \cup\{i, j_\ell\}) + v(T_{\ell-1}) \geq v(T_{\ell-1} \cup \{i\}) + v(T_{\ell-1} \cup \{j_\ell\})\\
\Longleftrightarrow \quad &v(T_{\ell-1} \cup\{i, j_\ell\}) - v(T_{\ell-1} \cup \{j_\ell\}) \geq v(T_{\ell-1} \cup \{i\}) - v(T_{\ell-1})\\
\Longleftrightarrow \quad &m_i(T_{\ell-1} \cup\{j_\ell\}) \geq m_i(T_{\ell-1})\\
\Longleftrightarrow \quad &\sum_{\ell=1}^k{m_i(T_{\ell-1} \cup\{j_\ell\})} \geq \sum_{\ell=1}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\ell=1}^{k-1}{m_i(T_{\ell-1} \cup\{j_\ell\})} + m_i(T_{k-1} \cup\{j_k\}) \geq m_i(T_0) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\tau=2}^{k}{m_i(T_{\tau-2} \cup\{j_{\tau-1}\})} + m_i((T \setminus \{j_k\}) \cup\{j_k\}) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\tau=2}^{k}{m_i(T_{\tau-1})} + m_i(T) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\
\Longleftrightarrow \quad & m_i(T) \geq m_i(S)
\end{split}
\label{eq:pvex}
\end{align}
The bottom line corresponds to C.3.
The final step C.3 $\Rightarrow$ C.1 is provided in Moulin (1988, p. 113).