Mathematical Micro/Macro Economics Textbook Recommendation I was formerly an economics major and now also majoring in mathematics.
I want a textbook that is rigorously based on mathematics; not just using mathematics whenever the author wants, but in a more unified way to explain a concept.
I've learned enough mathematics in undergraduate level so mathematics will hardly be a problem.
I've searched for information, but I really wanted textbooks that satisfy this specific condition; explaining concepts with sound mathematics in a unified way.

Any help is welcome.

You should also try Advanced Microeconomic Theory (3rd Edition) by (Jehle, Reny, 2011).
Note that, imho, Mas-colell-Whinston-Green (1995!) is the best choice for those with initial background in Math switching to Economics. When it comes to Economics majors the former seems to be more appropriate (and more modern).
For modern Macro with strong math see Introduction to Modern Economic Growth (Acemoglu, 2009).
To integrate the micro/macro problems with econometrics and math see Economic Modeling and Inference (Christensen & Kiefer, 2009). Hope, that helps.


LM curve from money demand Let $M^d (Y,r)=a+bY-cr$ where $M^d = M/P$ is the money demand in the economy.  $a,b,c>0$. Derive $LM$.
My try
$M/P=a + b Y - c r$
$b Y = -a + \frac{M}{P} + c r$
$Y = -\frac{a}{b} + \frac{M}{b P} + \frac{c r}{b}$
Is this all there is to it? Equal the equations and solve for $Y$?

The calculation is correct but the explanation is not simply 'equal the equations and solve for Y'.
Following Blanchard et al Macroeconomics: A European Perspective, 2nd ed. ch 5. by definition

LM curve represents all the combinations of output and interest rate for which the money market is in equilibrium, i.e. the demand for money equals the supply of money.

As given by your question the demand for money is given by $M/P$. Now in your question you do not explicitly label $a+bY−cr$  as a supply for money - but it looks like a supply for money. The reason why you equate the equations is because you want to equate supply and demand for money. If this would be given by some more complex relationship, for example you are given some extra equation for how $M$ or $P$ behaves, you would have to use that too and equating the two equations above would not be enough.
In addition, it is not necessarily to solve it for $Y$, you could as well solve it for $r$. In fact by convention in economics prices go on $y$-axis, so $r$ goes on $y$-axis and output $Y$ would go on $x$-axis, so if you would want to plot LM curve you would actually solve for $r$ not $Y$. This being said if you want to know what is effect of parameters and variables on output (which often is the case) it is not a mistake to solve it for $Y$ instead of $r$.


What is the term for the cannibalization of one tax by another? The context for the question is a situation where a government-owned lottery pays corporation tax, but an additional betting duty levied on individual gamblers reduces the lottery's profits, thereby decreasing the amount of corporation tax received by the State.

In this case we would say that duty levied on the lottery would erode the tax base for corporate tax. However, you should note that eroding tax base is a general term that can be applied to any action that decreases the taxable income/wealth/economic activity. But I dont think there is any more specific word in economic jargon just for that.


Event Study vs. Difference-in-Differences I am trying to perform Event Study methodology to analyse impact of group of events on group of stocks from certain sector. I noticed that I am getting surprisingly lot of positive results (very low p-value like ~0.00). When I sampled set random dates from my time period and run same Event Study analysis I got multiple low p-values again. Approximately 50%-60% per 100 random dates for each company. I assume that high volatility of my companies (and index) might one reason however I need confirmation here. But I have started wondering about different thing. I use same events for all companies from one sector. Not events specific for each company (like m&s for example). So I started wondering whether I should use Difference-in-Differences here? I heard that:
“Difference-in-Differences (DID) is more appropriate for systematic events that affect the whole market while event study is designed to examine impact of events specific for single company”
How true is that statement? As I said before in my analysis I examine impact of same set of events for each company like change in regulations for entire sector. Are there other alternatives beside Event Study and DID?

The statement is reasonably accurate. When we talk about the 'standard event study' in economics and financial literature (e.g. MacKinlay, 1997),  which based on your text  you are, they are not appropriate for analyzing single effects that affect whole market. There are several reasons for this:

A key assumption of event study is cross-sectional independence. A cross-sectional independence will be violated when in your sample multiple events happen at the same time. A textbook example would be stock listings occurring at the same day (See Brooks Introductory Econometrics for Finance). Such event clustering makes your estimates of test statistics and consequently $p$-values biased and as a result any inference from such event study will be most likely incorrect. If you are analyzing an impact of a single event you will have the most extreme case of event clustering as you are modeling the regulation as a separate event for every firm that happens exactly at the same period of time.
There are some solutions to event clustering. For example, you could solve it by not aggregating across the firms and just examining event impact on firm level do a summary analysis of all individual firm level studies.  An alternative would be to create a portfolio of firms  that experience the event at the same time and analyze the portfolio as a single firm (see again Brooks). However, in the first case the solution is not very elegant and it has its own problems that are beyond scope of this answer, but you could in principle do it. The second solution is not appropriate because in your case you would end up with a one big portfolio of all firms so you would end up with one aggregate observation.

Usually when you are interested in evaluating an effect of policy you do not want to just know if the policy had an effect on one day but if there are also effects that persist. For example, if you evaluate a policy like deworming pills in developed countries on student performance, you dont want to just know if there was one-off effect of this policy on student performance which lasted for few days or months, you would be interesting in knowing what was the effect long term. In such cases using event study is problematic as that would require you to use very long event window. However, this introduces another issues as event study is very sensitive to even small misspecifications if the event window is long (see Brooks again). This is why longer event studies use buy-and-hold abnormal returns (BHAR) instead of just cumulative abnormal returns (CAR) but if your event window is too long you will still have problems.

There are also further issues but the two I mentioned above are in my opinion the major ones when compared to DiD. Also yes there are other alternatives but I think that mentioning them all would be too broad. For example, you could model single event as a structural break or even as regime switch and analyze it that way. It will always depends on the specifics of the problem, for that it is always best to do very careful literature review on the topic and see how other people approach the problem and which issues they consider most important.


Economic examples of (sub)martingales A (discrete time) martingale is a stochastic process $\{X_t\}_{t\in\mathbb N}$ that satisfies, for all $t\in\mathbb N$,
$$
E(|X_t|)<\infty
$$
and
$$
E(X_{t+1}|X_1,\dots,X_t）=X_t.
$$
And a submartingale is one with $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$.

I know the classic example of the gambler's payoff in a fair game. But I'm hoping to find other real world examples of martingales and submartingales, particularly those pertaining to economics. Ideally, these examples would have some sort of empirical backing.

Applying the Law of Iterated Expectations on the defining property of a sub-martingale $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$ we have that

$$E\Big[E(X_{t+1}|X_1,\dots,X_t)\Big]=E(X_{t+1}) \geq E(X_t)$$

So a sub-martingale has also its unconditional expected value non-decreasing. Therefore any economic variable that appears to consistently increase in value is a candidate for being a sub-martingale, since the defining property here conditions only on the variable's past, not on any other factors that may influence it. National Output/GDP could be reasonably modeled as a sub-martingale, since it historically "always" increases, bar catastrophic events.

In general, any economic variable that can be modeled as a random walk with non-negative drift,

$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$

is a sub-martingale.


What are some real-world examples of the different types of trading blocs? The subsequent types of trading blocs have the following definitions:

Free Trade Area -- free trade among members
Customs Union -- free trade among members and common external tariff
Common Market -- free trade among members, common external tariff, and free movement of factors of production
Economic Union -- free trade among members, common external tariff, free movement of factors of production, and harmonization of all economic policies

What are some real-world examples of these different types of trading blocs?

There are actually five different types of trading blocs: preferential trading areas, free trade areas, customs unions, common markets and economic (and monetary) unions [1].

A preferential trade area (PTA) is a trading bloc that reduces tariffs on particular goods between two countries but does not abolish them completely.  There are no particularly well-known examples, but a list can be found here [2].

A free-trade area is the agreement among all member countries to eliminate tariffs, import quotas, and preferences on goods and services traded between each other (the free-trade as defined in the question) [3].  Probably the best known free-trade area is NAFTA; but others are listed here and here.

Again, there are no well-known customs unions, but a list can be found here.

A common market has free trade and free movement of the factors of production but other economic policies of the countries may be different (one of the differences between a common market and an economic union).  You may also want to look at single market and unified market.  There are no well-known common markets, but the previous link also contains a list.

The best known economic union is the EU though more can be found here.  An economic and monetary union has the same qualities of an economic union but each member nation also uses the same currency.  One example would be the nations in the EU who also utilize the Euro.


What happened to the national debts after Bretton Woods When Bretton Woods was created and the world went on to the US Dollar standard, what happened to all the national debt's of all the countries like Japan, Britain, etc?

The Bretton Woods system was a fixed exchange rate regime, but local currencies existed. Debt was unaffected (other than any debt repudiations that may have happened after the end of World War II).


Comparative Advantage Question The theory of Comparative Advantage in production and trade specifies that countries should engage in the production of goods and services for which they have a comparative advantage or lower opportunity cost than others. Do you think this theory can be effectively applied even where there is a wide technological or developmental gap between two countries?

Yes it can be used. The theory applies regardless of what’s the source of comparative advantage.

For example, if country A is able to produce 200 widgets and 50 apples and country B is able to produce 5 widgets and 45 apples you get comparative advantages - A in widgets and B in apples - and it does not really matter if these are due to technology, economic development, work ethic etc.

As a matter of fact differences in technology are sometimes mentioned in undergraduate class as source for the comparative advantage.

Actually in the original example of comparative advantage used by Ricardo the example was of England that had comparative advantage in cloth production due to new industrial technologies and Portugal which was more rural and hence had comparative advantage in wine. Also at the time England was more economically developed than Portugal- at least to the extent that can be seen from historical estimates provided by Maddison.


If I don't pay a debt, then the creditor takes my goods. Why, then, do Greek creditors not take Greece? Normally, when you don't pay a debt, your creditors take your goods (house, car, etc).

If Greece cannot pay its debt, can its creditors take Greek goods (structures, cities, industries, lands, etc)? Can Germany, or other organizations, become owners of Greece?

Maybe that's a silly question, but for a non-expert it sounds logical.

In general, there are three kinds of debt:

Secured debt, like a mortgage or a repurchase agreement. With a mortgage, for example, the debt is secured by a lien on the home, and if the debtor does not pay, the creditor can seize the home.
Unsecured debt, like a credit card or corporate bond. Governments will generally allow creditors to liquidate many of the assets of an insolvent individual or firm to meet unsecured obligations, subject to (often significant) limits. For individuals, things like retirement accounts and primary residences are often off-limits for debt collection.
Sovereign debt. Sovereigns typically issue debt under their own law, and countries generally don't set up their legal systems in such a way as to allow other countries to take their things. However, when sovereigns issue debt under foreign countries' law, as Greece has done with some UK-law bonds, payouts tend to be higher, and overseas property is in fact occasionally seized. However, seizing the domestic property of a defaulted sovereign is effectively impossible, as one of the benefits of being a sovereign is that you control your own territory— attempting to seize another nation's "structures, cities, industries, lands, etc." (setting aside the fact that most structures, cities, industries, and land in Greece are not owned by the government) is what is known as an act of war.


Complementary slackness conditions (Kuhn-Tucker) Consider the problem of maximising a smooth function subject to the inequality constraint that $g(x) \leq b$. The complementary slackness condition says that

$$ \lambda[g(x) - b] = 0$$

It is often pointed out that, if the constraint is slack at the optimum (i.e. $g(x^*) < b$), then this condition tells us that the multiplier $\lambda = 0$. I agree with this. However, it has also been said that, if the constraint 'binds' (which implies that $g(x^*) - b = 0$), we must have $\lambda > 0$. Is this true? As a logical matter, it is not immediately implied by the complimentary slackness condition: we could have both $g(x^*) - b = 0$ and also $\lambda = 0$.

Edit: it has been demonstrated here why we can have both $\lambda = 0$ and $g(x^*) - b = 0$ (thanks to @markleeds for the pointer). I am wondering, however, whether we can have $\lambda = 0$ while the constraint also binds (i.e. makes a difference to the solution -- note that this is subtly different from the constraint holding with equality). I suspect that the answer is 'no' given that $\lambda$ reflects the effect of slightly relaxing the constraint on the objective function. However, I would appreciate confirmation of this.

It is possible to have
$$g(x^*) = b\; {\rm and}\; \lambda^* = 0$$.
When the multiplier is zero and the constraint is equal to zero, then
a) The constraint does not really "bind"
b) That's why the multiplier is zero.
What does it mean "the constraint does not really bind"?
It means that the solution $x^*$, that makes $g(x^*) = b$, would be chosen even if the constraint was not imposed. In that sense, the constraint is not really binding, because it does not really forbid us to go where we wanted to be, because we are already there.
Consider the simple example
$$\max_x \{-ax^2 + bx\},\qquad s.t. \;x \geq \frac{b}{2a} $$
The Lagrangean is
$$\Lambda = -ax^2 + bx + \lambda \left(x -\frac{b}{2a}\right)$$
and the f.o.c is
$$x = \frac{b+\lambda}{2a}$$.
Try cases:
a) $\lambda^* = 0$ leads to $x^* = \frac{b}{2a}$, which is the unconstrained f.o.c. also.
b) $\lambda^* > 0$ the f.o.c. initially indicates that $x^* > b/2a$. But then the constraint is not binding and we should have $\lambda^* =0$ : contradiction.
So we see that in this case the solution is
$$x^* = \frac{b}{2a},\;\;\; \lambda^* = 0.$$
So the constraint appears to be binding, but it really is not.


Effects on transit system shutdown based on local workforce end-of-day times? I'm curious if there are any studies based on logistics and economics of shutting down the bus/transit system before the general population's working class can go home.

For example, the small town I live in, yet large enough to have transit, a huge retirement community (might factor in), and some amenities like app-based shopping (but no Lyft type service) has a bus system that will stop at 8:30 within a 3 mile radius of the main hub, 6:30 for farther out east, and about 7:30 for west (basically early). The stores, however, close at 9 or 11, as do most restaurants.

I began to factor in a few things, and have been web-searching to see if there are any studies.

New workforce (teenagers) probably haven't saved enough for a "good" car, which I would guess encourages an older, more experienced workforce. A commuter can not shop after work (no time) nor can they work until this time. The meaningful fact is these are minimum-wage jobs, meaning that the ones most likely to need the bus after 9 are the ones who work at the stores.

Supply and demand effects, one would assume that younger employees work mornings and mid-day, and the later ones find rides through friends. Thus, the pizza/party and taxi economy thrives, but there is an unfulfilled demand for low-experienced workers in late shifts.

I am just proving I have done some thought-searching on this, as the keywords to search for, if a paper exists, tend to get me transit schedules and not studies on such.

That's a fascinating idea, and I'd thoroughly suggest additional research into the matter (especially because you might be able to conduct a quasi-experimental analysis using instances of changing public transportation schedules on labor market factors, such as Boston's temporary decision to keep the MBTA rapid transit open later).

As far as your interest in other academic studies, I'd ask for a bit more clarification for a complete answer: what is the outcome variable(s) you are most interested in? Are you thinking about the impacts on the labor market (the distortionary effect on which groups are able to take jobs in the later shifts of retail, for example)? Or are you more interested in the potential for economic growth if hours were extended?

If you clarify the above a bit more, I can try to search for more resources. However based on what you've mentioned, there are a few academic studies I found that might be of interest:

This Study by Fan, Guthrie and Levinson looks at the impact of light rail service on labor market outcomes for various types of workers. They cite two other articles (here and here) that also look into the impact of public transportation on labor market outcomes for different groups of workers. There are other citations in all three documents that you might find interesting along that dimensions.

This article by Guthrie and Fan Examines the impact of public buses on economic development in the Twin Cities, Minnesota. This study is one conducted by the American Public Transportation Association (so I might take some of their findings with a dose of skepticism) and details a variety of the economic impacts public transportation investment can have in the surrounding area.

There are a few other studies that are more tangentially related to what seems to be your central question, but still might be interesting. This article, by Bauernschuster, Hener and Rainer looks at the impact of public transportation strikes on traffic, health outcomes, pollution, and other factors. The impact on traffic is also examined in this NBER article by Anderson.

It may also be worthwhile (again, not entirely sure what your main research question is on specifically) to look outside some of the normal "econ research channels." This study in PubMed looks at the impact of Strikes in London's Underground on bicycle usage.

I hope some of those helped. If I can refine my answer further to better assist, just let me know!!


How to measure Tobin's q? How do people calculate Tobin'q empirically?

Does anyone use it to make investment decisions? In particular, the version I learnt has no company debt involved with only capital under consideration, will this affect empirical research?

If you are interested in Tobin's (average) q,  you might find useful A Simple Approximation of Tobin's q (Chung and Pruitt (1994)). That paper has a highly accurate approximation for Tobin's Q using Compustat. While the JSTOR copy I linked to is pay-walled, other versions appear not to be.


Is the international economic system too chaotic to "work towards" carbon emission reduction I often get frustrated when I read calls to action on climate change.   It goes something like this: "Solutions are within reach - but we must act now!"

If one assumes that the international economic system is chaotic and reactive,  and that no one is in charge, where is the theory to back up what seems to be a wild claim that "we can act".  Who are "we", and are markets even able to deal with information about the common good or the future.

Suggestions for further reading on relevant theory would be appreciated.

There are already precedents that show the international capacity is there, technically, and economically to deliver the carbon reduction necessary, in time. And there are indications that the political will to do so is growing to sufficient levels, too.

Firstly, with the number of participating nations being 195 (the UNFCCC nation parties), that's about the right size to create a management group that prevents the destruction of the commons, despite the short-run market incentives being ones that would otherwise lead to such destruction: see the Nobel-winning work by Elinor Ostrom for lots of information on this - I highly recommend it, it's a very good read. This addresses the aspect of your question about whether the market can accommodate information about the common good and the future.

Secondly, there are already examples of individual nations putting primary legislation in place, and acting on it: the UK's Climate Change Act was the first such law in a major industrialised nation; other countries have done similar.

Thirdly, there are already examples of nations collaborating on global environmental problems successfully: the international action to reduce emissions of ozone-destroying chemicals, the Montreal Protocol, is working.

And fourthly, the COP21 Paris Agreement showed unanimity among the 195 nations plus the institution of the EU in the need to address this problem. Pause for a moment to consider the enormity of that. The Paris Agreement wasn't just approved by majority consensus, but unanimously by all 195 countries.

All of these things happened within the context of the international economic system. So it's certainly not a barrier.

And we know that the economic incentives to prevent catastrophic climate change far outweigh the costs of doing so, at any sane discount rate. However, there is a principal-agent problem, and decades of lag between cause and full outcome; therefore market mechanisms alone cannot be sufficient. But there are no pure market economies - it's all a blend of private and public agents - so that's only a hindrance, but not a barrier.


Nash social welfare function with negative exponential utilities I got one question about the Nash-SWF. Typically it is defined as the product of individual utilities, ie.
$$
NSWF:=u_1(x_1) \cdot u_2(x_2) \cdot u_3(x_3) \cdot ...
$$
For this to make sense, individual utilities are restricted to always being positive. Is there a way to adjust the Nash-SWF to work for utility fcts that are always negative, like $-e^{-ax}$? Meaning all individuals have the same utility fct. which is $-e^{-ax}$.

Thanks a lot!

I completely agree with @FooBar. Of course the utility function can be negative in some of its range. Conceptually, one could invoke the cardinality vs. the ordinality argument as a response to your question. I would think that your problem is as follows:

$$
\
max\, SWF=u(x_{1})\times u(x_{2})\times...\times u(x_{N})
\
 s.t
\
f(x_{1}....x_{N})\leq g
\
$$

The constraint(s) can be some resource constraint. All that matters here is that the SWF satisfies the normal properties associated with the existence of a maximum on the constraint set (You can invoke the assumptions of the Weierstrass Theorem here). Moreover, you should check Second Order Sufficient Conditions (in this case the negative definiteness of the Bordered Hessian) for a maximum. The value of the welfare function by itself is irrelevant as long as it is a maximum on the constraint set.


Must a market-based economy constantly expand? I vaguely recall from my reading of Adam Smith’s “The Wealth of Nations”, during college, that he opined that a market-based economy must constantly expand.  Is this recollection correct and, if so, what is the specific quote/reasoning?  Is it not possible for such an economy to be static and self-contained?

No idea who said that first, but I have seen similar assertions in economic commentary many times. The logic was never fully explained, but revolved around the notion that we need compunding income to pay compund interest. Authors treated the idea as so obvious or well known that they never saw the need to explain the details, or cite earlier authors. (It’s been awhile since I’ve seen the assertion, so I cannot give examples.)

In theory, the belief that the economy must expand appears incorrect. One could look at the various stock-flow consistent (SFC) models in “Monetary Economics” by Wynne Godley and Marc Lavoie, and one sees that steady solutions are easy to find. Although debt compounds, borrowers just pay down debt.

As a practical matter, the tendency is to prefer growth. However, in a services based economy, it is unclear whether real GDP growth gas to correspond to greater natural resource use, which creates a debate whether it is “real growth” or “paper growth.”


Transversality Condition in neoclassical growth model In the neo-classical growth model there is the following transversality condition:

$$\lim_{t\rightarrow\infty}\beta^{t}u'(c_{t})k_{t+1}= 0,$$
where $k_{t+1}$ is the capital at period $t$.

My questions are:

How we derive this condition?
Why do we require this, if we want to rule out paths with no debt accumulation?
Why are the Lagrange multipliers $\beta^{t}u'(c_{t}) = \beta^{t}\lambda_{t}$
the present discounted value of the capital?

The transversality condition may be more easily understood if we start from a problem with finite horizon.

In the standard version, our objective is to
$$
\max_{\{c_t,k_{t+1}\}_{t=0}^T} \sum_{t=0}^T\beta^t u(c_t)
$$
subject to
$$
\begin{aligned}
f(k_t)-c_t-k_{t+1}&\ge0,\quad t=0,\dots,T &&\text{(resource/budget constraint)}\\
c_t,k_{t+1}&\ge0,\quad t=0,\dots,T &&\text{(non-negativity constraint)}
\end{aligned}
$$
with $k_0$ given. The associated Lagrangian (with multipliers $\lambda_t$, $\mu_t$, and $\omega_t$) is
$$
\max_{\{c_t,k_{t+1},\lambda_t,\mu_t,\omega_t\}_{t=0}^T}
\sum_{t=0}^T \beta^tu(c_t)+\lambda_t(f(k_t)-c_t-k_{t+1})+\mu_tc_t+\omega_tk_{t+1}
$$
The FOCs are
$$
\begin{align}
c_t:&& \beta^tu'(c_t)-\lambda_t+\mu_t&=0,\quad t=0,\dots,T \\
k_{t+1}:&& -\lambda_t+\lambda_{t+1}f'(k_{t+1})+\omega_t&=0,\quad t=0,\dots,T-1 \\
k_{T+1}:&& -\lambda_T+\omega_T&=0,\quad T+1 \tag{1}
\end{align}
$$
with the Kuhn-Tucker complementary slackness conditions: for $t=0,\dots,T$,
$$
\begin{align}
\lambda_t(f(k_t)-c_t-k_{t+1})&=0 & \lambda_t&\ge0 \\
\mu_tc_t&=0 & \mu_t&\ge0\\
\omega_tk_{t+1}&=0&\omega_t&\ge0\tag{2}
\end{align}
$$
Since resource constraint must be binding in all periods, i.e. $\lambda_t>0$ for all $t$, it follows that at the last period $T$, $\omega_T=\lambda_T>0$, which in turn implies $k_{T+1}=0$.

Usually we assume $c_t>0$ for all $t$ (the Inada condition), and this implies $\mu_t=0$ for all $t$. So the consumption FOC becomes
$$
\beta^tu'(c_t)=\lambda_t \tag{3}
$$

Looking at conditions $(1)$ $(2)$ and $(3)$ in the last period $T$, we get
$$\beta^Tu'(c_T)k_{T+1}=0$$
Extending this to the infinite horizon, we get the transversality condition
$$\lim_{T\to\infty}\beta^Tu'(c_T)k_{T+1}=0$$

The intuition of the transversality condition is partly that "there is no savings in the last period". But as there is no "last period" in an infinite horizon environment, we take the limit as time goes to infinity.


Why did the hybrid Phillips Curve become flatter (as empirical data shows) after the crisis? The Phillips Curve equation is:
$$\pi_t=\pi_t^e + \beta_1 (\beta_2 Y_t + (1-\beta_2) Z_t) + \varepsilon_t^\pi$$
where $\pi_t$ is the inflation, $\pi_t^e$ is the expected inflation, $Y_t$ is the output level, $Z_t$ is the real exchange rate, and $\varepsilon_t^\pi$ is a stochastic shock.

In a small open economy (semi-structural framework), the impulse response functions to a domestic demand shock are the following:

After the crisis, the Phillips Curve became flatter. My question is, how can one interpret the change in the slope of the Phillips Curve after the crisis?

(1) The Phillips curve is not a structural thing, it is not surprising that the relationship between unemployment and inflation is not constant over time.

(2) There's a nice AER paper by Simon Gilchrist and coauthors that argues that inflation did not go down as much as in earlier recessions because financially constrained firms needed to keep their cash-flows up and increased prices.


What happens in a bartering system when a resource is plentiful but cannot be paid for? Here is a hypothetical scenario:

Let's say there are two tribes which barter goods. Tribe A has an fairly large amount of fish, but no bricks. Tribe B has an extreme abundance of bricks, but no water in their land to go fishing.Tribe A trades 5 fish with Tribe B for 100 bricks.  This arrangement exists peacefully for many years.

At some point Tribe A creates tools to create bricks and no longer requires the bricks of Tribe B. Tribe B still requires the fish of Tribe A but now has nothing to pay for the fish.

What would happen in this instance? Is there historical evidence in which a similar scenario has occurred? Is there an economic theory that explains what will happen?

Tribe A will not necessarily sever barter with tribe B. This has to do with Production Possibility Frontier and related concepts such as specialization & utility.

Tribe A's production of bricks could certainly press tribe B to agree to a bricks/fish exchange rate that is more convenient to tribe A than it was prior to the latter's invention of tools to create bricks.

In an extreme scenario, tribe A's bricks might be of better quality and much cheaper than tribe B's bricks. That might render A unlikelier to maintain barter with tribe B. If so, tribe B might want to consider fish substitutes such as hunting, veganism, veganism's closest alternative (i.e. cannibalism), acuphagia (of A's new tools), or outright eating bricks.


How does the money supply behave when bank loans are repaid? In a fractional reserve system when banks lend out money, that money is created out of thin air by a accounting journal entry, and the money supply goes up by the amount of the loan & when the loan gets paid off, that money disappears back into thin air and the money supply goes back down which is often also described as "destroying the money"

As under normally amortized loan, out of monthly payments, some part goes towards interest & some towards principal repayment, so the question is does the money equivalent of principal repayment disappears into thin air every time the payment is made towards principal of the loan, or does it only disappears at end of the loan term all at once when the loan has been repaid in full?

If the money gets destroyed or rather disappears from the money cycle every time the principal is repaid from the monthly payments, then does it means that the banks are only left with the interest parts & nothing out of the principal portion on their books?

Also how does the money gets depleted from the money cycle and disappears into thin air in the case of "interest only loans", where during the term of the loan only interest payments are being paid by the borrower & the principal is repaid all at once at the end of the term of loan in form of a single one time payment.

So is it that in such interest only loans because the principal gets repaid all at once at the end of the loan term so at that time immediately it gets depleted from the money cycle all at once & disappears into the thin air?

And most importantly: if all of the money equivalent of the principal amount of loans disappears then how does the banks end up recovering the actual amount loaned by them to the others & not just the interest on that loan?

The money is removed when the loan principal is repaid. The actual point in the loan this occurs depends on the loan terms. For a typical compound interest rate loan, this means a small portion of the principal is repaid every month, and a matching liability deposit (money in the customers account) is removed.

For an interest only loan, this occurs at the end of the loan - assuming all the principal is repaid then.

Interest payments essentially circulate through the monetary system. They're deducted from the customer's account, recognised as income by the bank, and then paid out as some form of expense, e.g. salaries, rent, taxes, etc. They may also be moved into an internal account to provide required loss provisions on loans.

Loan defaults are also an expense, and this is the achilles heel of the banking system. If defaults on loans exceed the bank's loss provisions and profits from interest, then the bank will have to write off against its capital - and this interferes with the regulatory controls on lending, causing the money supply to shrink. In practice, central bank or government intervention is inevitable if this occurs.

As far as the relationship of principal to the money supply. Essentially the banking system relies on new lending always being sufficient to replace the money being removed. In most banking systems, new lending is typically in excess of loan repayment, and so we see the money supply more or less continuously expanding.


Is it possible for a firm does not have shutdown point? If a firm has a quadratic cost function, is it means that the firm does not have shut down point and never shut down? Are the changes in fixed cost do not affect the firm's shutdown decision?

$TC(q)=10+3q+0.5q^2$ is a quadratic cost function and has a shutdown point at $P=3$.


Would a world-wide currency be a good idea? While every country has it's own political ideals, enforcing laws and rules, in an ever more globalized world, I believe the idea of fragmented economy is limiting big growth opportunities while creating a lot of unnecessary problems and bureaucracy around the world. So would changing to a globalized currency model be a good idea?

The model

A single currency would require an entiity whose would control it and be responsible for it. While we could create a new one, we might as well just use an already existing entity to be responsible for it, like the United Nations. Picking the UN to regulate this currency world wide (or at least to the big list of State nations that are part of it), would give us the opportunity to add an extra layer to the government layers on top of the Nation layer, similar to what the European Union does.

This would allow the United Nations not only to suggest rules, like the Human Rights, but to enforce them as law to its member states.

The need

As mentioned before, we are living in a world more globalized every day. Some people work to different countries other than the ones they live in through the internet. This creates a currency exchange problem, where he is removing currency power from the country he's working for and giving it to the country he lives in, effectivelly lowering the first's value and raising the second's value. A single currency could solve this problem and also getting out of the way the whole exchange problem from the middle, and in some cases, the international bank transfer bureaucracy.

A single currency with an enforing entity controling entity could use this power to help countries in the desperate need to grow, I'm thinking about places like Zimbabwe, Niger and Ethiopia. By retrieving in the form of taxes a percentage of each nation's Gross Domestic Product (let's say, 3%?), summing it all up, then averaging it by the number of member states, and then redistributing this to each country, we would be in fact spreading resources from wealthy parts of the world to the poorest in a fair way. Everyone contributes as they can so a few gets back. To avoid countries which could use this system to just sit back doing nothing, waiting for the money to come in, their governments are obligated to use this money to invest in the country's industries (or whatever are the country's activities which make money)

Of course, this is just a simple idea which I didn't develop, it can have much more potential and bring it's own problems, that's why I'm posting it here, so we can discuss about them. Also I'm sorry for my weird english and terms used, it's not my main language and I'm not an economics student, but I hope it's understandable.

I agree with the members that have voted to close, but I felt obliged to point out an important confusion in the question. The OP writes

  "A single currency with an enforcing entity controlling entity could use this power to help countries in the desperate need to grow,... By
  retrieving in the form of taxes a percentage of each nation's Gross
  Domestic Product... and then redistributing this to each country, we
  would be in fact spreading resources from wealthy parts of the world
  to the poorest in a fair way."

It appears therefore that the OP focuses on issues of global income/wealth inequality, and he thinks that, under a common currency, redistribution would be easier...

...The OP fails to see that, in order to have a global currency, it would take a "geopolitical will" from the parties involved, which, if it existed, the contribution of a common currency as a "redistribution facilitator" would be negligible.

One important reason of why we do not have a global common currency, has to do with the fact that we do not feel that much "connected", truly "parts of a single whole". Hence, wealth redistribution to the extent imagined by the OP is not something universally desired. Because if such a sense of being part of a single whole was the dominant flavor of human existence, the desire to redistribute would flow rather naturally from it, and then it could actually happen whether we had a common currency or not.

Now I am voting to close also.


Comparative statics with multiple equations using calculus I am currently learning a three market model of the economy, with the goods market, money market and factor market being in equilibrium respectively when-
$$Y=C(Y-tY)+ I(r) + (\bar G) + \bar{NX}$$
$$\frac{\bar M}{P} = M^d(r, Y)$$
$$P=P^e f(\frac{Y}{\bar L}, \bar \mu, \bar z)$$
Here all the variables with bars above them are exogenous, and the rest are endogenous. I have been taught to trace the effect of the change in a parameter by using graphs. The first equation and the second give us the IS and LM curves respectively in the $(Y, r)$ space, they together give rise to the downward sloping AD curve in the $(Y, P)$ space. The third equation generates the upward sloping AS curve in the same space.
For example, increasing G shifts the IS curve to the right, as given an interest rate, Y rises. This in turn raises r and Y. The increased Y at any given price level shifts the AD curve to the right. This raises P and Y again. The original LM curve moves slightly left but not to completely offset the original increase in Y, as the AD-AS diagram makes sure that Y actually increases.
I often convert the diagrammatic description to calculus and for each specific market, assuming all other markets to be not affected, it seemed fairly easy.
But I got completely lost when I tried to account for derivatives such that all the three equations are simultaneously satisfied, that is, to translate the diagrammatic logic into calculus.
How should I go about it? Also, where can I learn that technique for any arbitrary set of equations, and not just this particular model, to be able to use this in other models in the future?

Since this looks like a homework and there was no attempt made, in order to avoid giving full solution I will just give you conceptual explanation using equation 1 and 2 (you can later add equation 3 using steps below).
The standard method how you approach this is by taking total derivative and solving the system for $dY$. Total derivative is similar to regular derivative but you will take it with respect to every variable in the system.  Here you will have set of equations:
$$ dY = C_Y'dY + C_t'dY + I_r'dr + dG + dNX \\
 \frac{PdM-MdP}{P^2}= M^{d'}_Y dY +  M^{d'}_r dr $$
The same logic can be also applied to equation 3 that I purposefully omitted.
Now you can employ any method for solving system of multiple equations you want. The most common methods of solving sytems of equations are substitution, linear combination method (i.e. the method where you add multiple equations), various matrix methods such as Cramer's rule. However, no matter what method you use they will all work but get progressively more difficult the more equations you have. There is really no good shortcut here. Even matrix methods that work reasonably well when you have multiple equations can quickly become unwieldy as number of equations grow. In practice you would not solve difficult systems by hand but just set up the system in pyhton or matlab and let it solve it for you (however, in classroom setting it is unlikely you will encounter systems with more than 4 equations).
In this case the substitution is the quickest method so I will us that one. We can solve the equation 2 for $dr$:
$$ dr = \frac{PdM-MdP}{P^2M^{d'}_r} - \frac{M^{d'}_Y}{M^{d'}_r} dY  $$
Now substitute this equation back into the first one to get:
$$dY = C_Y'dY + C_t'dY + I_r' \left( \frac{PdM-MdP}{P^2M^{d'}_r} - \frac{M^{d'}_Y}{M^{d'}_r} dY \right) + dG + dNX$$
solve for $dY$:
$$ dY =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} \left( I_r'  \frac{PdM-MdP}{P^2M^{d'}_r} + dG + dNX\right)$$
This expression should already be familiar to you from your macro 101 class - the first fraction is the multiplier and the second brackets are the exogenous sources of spending.
Now if you want to examine what happens when let's say government spending increases ceteris paribus you will just set all other $dX=0$ save for $dY$ and $dG$ (because other things will not change) giving you in this case:
$$ dY =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} dG $$
And then we can find the partial derivative of $\frac{\partial Y}{\partial G}$ simply as:
$$ \frac{\partial Y}{\partial G} =  \frac{1}{\left(1 -  C_Y' -  C_t' - I_r' \frac{M^{d'}_Y}{M^{d'}_r} \right)} $$.
And you are done this expression tells you by how much the $Y$ changes when $G$ changes. You can redo the above for any other variable you want, and you can easily add the third equation as well (just take its total derivative and do 1 extra substitution that is required). The above method will work for arbitrary system. Sometimes it is easier to use different method of solving the system of equations but that is very situational and there I don't have better advice than just to practice various ways of solving systems and in the end you will develop 'knack' for knowing which method is quickest in particular case (although they will all work just some are sometimes quicker/easier to apply).


What would a negative externality diagram look like if the initial price is equal to 0 and then increased with tax? The price of plastic bags in Scotland has been increased from free to 0.05 pounds sterling. How would I represent the negative externality decrease using a diagram in such a situation?

This is what I have so far. I'm not sure that it is correct.

Prior to the tax, the marginal private cost is zero (so the MPC curve is a horizontal line along the x axis).

The individual chooses to consume where marginal private benefit is equal to marginal private cost (i.e., where the MPB curve crosses the x axis).

A tax of 0.05 implies that the marginal private cost (i.e. the cost per unit consumed) is now 0.05. This means that we get a new MPC curve that is a horizontal line at p=0.05. The new privately optimal consumption occurs where this new MPC curve intersects the MPB curve, which results in a lower quantity consumed.


Quasi concavity of utility function I am looking for a method to prove that the following function is quasi-concave in $\alpha$  (or find conditions under which it is true):

$ \pi=F(-k)(f(0)^2-f(h(1-\alpha))^2)+ \frac{1}{2}-\frac{1}{2}f(0)^2-\frac{1}{2}f(-k^2)(2F(h(1-\alpha))-1)^2 $,

where:

$F$ - is a CDF function of Normal Distribution, $N[0,\sigma^2]$;

$f$ - is a PDF function of Normal Distribution, $N[0,\sigma^2]$;

$k$ - exogenous patameter, k $\in [0,+∞)$;

$h$ - exogenous patameter, h $\in [0,+∞)$.

Any help would be appreciated.

UPD

I have forgotten to mention that I am trying to prove quasi-concavity on the interval $\alpha \in [0,1]$

Your conjecture seems to be contradicted, at least for small values of $\sigma$.
You can draw the function with the following R-code:

qq_f = function(x,k,h,sig){
-pnorm(-k, sd=sig)*( (dnorm(h*(1-x), sd=sig))^2 ) - 0.5*dnorm(-k^2, sd=sig)*( 2*pnorm(h*(1-x), sd=sig) -1 )^2
}
curve(qq_f(x,k=0,h=1,sig=0.5),col='blue',xlim=c(-1,3),type='l',main="A candidate quasi-concave function")


Internal rate of return Trying to finding the internal rate of return R:

Net present value of costs is $\ Y(s)$

Net benefit is $\ Y(s+1)-Y(s)$ in every period t up to$\ \infty $

The answer is given as $\ R=\frac{Y(s+1)-Y(s)}{Y(s)} $
Is this the correct approach (NPV = 0)?:
$\ Y(s) = \sum_{t=1}^{\infty}\frac{Y(s+1)-Y(s)}{(1+R)^t} $

Assuming the present is taken to be period 0, your approach is correct, subject to one caveat.
As you suggest, the internal rate of return is the discount rate at which the discounted present values of all the costs and all the benefits of a project sum to zero.  In this case, you are given the stream of benefits, and the right hand side of your equation correctly represents the discounted present value of that stream.
So far as the left hand side is concerned, instead of being given a stream of costs to treat in the same manner, you are just given the net present value of costs.  If the question is from a textbook and that is all the information given, you can hardly do anything else than use that net present value.
The caveat is that it's unclear how that net present value of the costs has been obtained. If it was calculated using any discount rate other than the true internal rate of return, eg an organisation's standard discount rate for evaluating the net present value of proposed projects, then (except by coincidence) your approach will not give the correct internal rate of return (and the given answer will be wrong).


Use Lag Operator to find Lifetime Budget Constraint The budget constraint is

$c_t + \tau_t + s_{t+1} =w_t(1-l_t) +(1+r_t)s_t$

And assume

$\underset{t \longrightarrow \infty}{lim} \ \displaystyle{\frac{s_t}{\Pi_{i=1}^{t-1} (1+r_i)}} = 0$

Lag Operator $L$ is defined as $L \cdot x_{t+1} = x_t$

How can I get lifetime budget constraint using the Lag Operator?

Many Thanks!

@T. G.: I think I obtained the expression for $s_{t+1}$ as a function of the other variables. I don't know what lifetime budget really means so I'll write the answer here and hope it's useful.

$s_{t+1} = \sum_{t=0}^{\infty} \lambda_{t} (w_{t}(1-l_{t}) -c_{t} - \tau_{t})$

where $\lambda_{t} = \prod_{i=0}^{t-1} (1 + r_{t-i})$.

I'm not certain if it's correct but intuitively, it looks like exponential smoothing of some kind of expression each period, except that the smoothing constant is not constant and is a function of the past interest rates ??


Why does local non satiation imply the constraint is binding? Local non satiation says that for any $x \in X$ and $\epsilon > 0$, there exists $y \in X$ such that $d(x,y) < \epsilon$ and $U(x) < U(y)$.

I don't understand why this implies that $px^* = m$ if $x^*$ sovles consumer problem. If we think of $x \in R^2$, it implies that you can find a $y$ that is strictly preferred in the small neighborhood of $x$. In that case, even $x$ is on $px = m$, LNS seems to imply that there is a $y$ that is strictly preferred than $x$, and that $y$ may not be on the boundary since LNS only says there's a increasing direction but doesn't say which direction it is increasing.

The OP is correct in pointing out that "Local Non Satiation (LNS) only says there's a (utility) increasing direction but doesn't say which direction it is increasing". Namely, we entertain the possibility on dealing with "bads" also, not only with "goods". MWG Microeconomic Theory book page 43 Figure 3.B.1 depicts exactly such a situation.

But it is the case that, when the bundle set is $\mathbb R_+$, under LNS not all items can be bads. Because then, the zero vector will be a point of satiation (and so it would violate the LNS assumption).

So using non-negative quantities of items and imposing LNS forces us to consider only the cases where at least one item in the bundle is a good and not a bad, in which case "more is better" for this item.

Then, we can prove that local non-satiation implies exhaustion of the available budget.

Ad absurdum, assume that $px^* < m$. Under LNS for every $\epsilon >0$, there exists a $y(\epsilon)$ that is more preferred to $x^*$. If some $y(\epsilon)$ is feasible, $py(\epsilon) \leq m$, then $x^*$ cannot be the optimal choice in the first place.

So the question is : Is it possible that all $y(\epsilon)$ that are preferred to $x^*$ under LNS, are infeasible, $py(\epsilon)>m,\;\; \forall \epsilon>0$?

I guess the OP can take it from here.


A hypothetical question on neutrality of money Let's assume that we start with an economy producing two goods, A and B, with each priced $100. Each good is weighted using the same quantity, so CPI equals 100. Nominal wage is also 100 dollars. An increase in money supply cause an equal increase of 20% for both good A and good B, while real wage fall because that CPI now equals 120. The question is that, relative prices between A and B did not change while the real wage fell (against CPI), what would be the result of these on real GDP? I presume that the producers would want produce the same amount as their relative prices did not change while the workers would reduce their quantity supplied as their real wage fell, compared to CPI. Thus, this would bid up wages until equilibrium in labor market is reached, as firms would find it non-optimal supplying any other quantity than initial level of real GDP because their MC would be bigger than their MR or vice versa.

Did I got the situation right? Any thoughts? Thanks in advance.

I think I figured out the answer using Excel. I assumed that there was a 100% inflation in goods market so that the prices of two goods doubled, while keeping the nominal wage fixed. The result was that an individual producer would still be making positive inflation adjusted profits. So I assume that in the scenario I described above, real GDP would still increase even though relative prices in goods market stays the same because that costs adjusted for inflation fall. When I also doubled the nominal wage so that the equilibrium in labor market is back to its initial level, profits were down back to the initial levels.

Here are the screenshots from Excel

A hypothetical firm and its cost curve.

Real price and real cost of the firm after a 100% inflation in the goods market.

Inflation adjusted profits of the firm after doubling the nominal wage from 120 dollars to 140 dollars. Profit levels are returned to their initial amounts and I presume that at this point firms would go back to produce the initial amount of real GDP.

I think what I got incorrect was that I was trying to compute real wages by dividing it to a ratio of nominal prices to CPI using the formula (W)/(P/P*) where P* representing CPI level. The correct usage should be (W/P*)/(P/P*) so that an equal increase in price of an individual good and CPI while holding the nominal wage fixed should result in a lower real wage, inducing higher profits than the initial level of prices and costs thus resulting in more labor demanded by the firm producing good A.

When I doubled the nominal wage and used the (W/P*)/(P/P*) formula, real wage went back to the initial level so that there was no profit opportunity so the firms reduced the extra real GDP supplied and therefore went back to their original levels of production.


OLS, Frisch-Waugh-Lovell Theorem and few observations I was talking to a colleague a few days ago and the following question came up:

Why can't we just use projection matrices and the FWL Theorem to "gain" a few degrees of freedom in case of very few observations?

Imagine a case where you want to estimate

$$y_t = \beta_0 + z\beta_1 + x_2\beta_2 + x_3\beta_3 + u_t$$

but unfortunately, you just have 5 observations (extreme example). Luckily, you are just interested in $\beta_1$. Now you could just create some projection matrices in the form

$$ M_x = I - x'(x'x)^{-1}x'$$

and project off your "unwanted" $x_2$ and $x_3$ (following Davidson and MacKinnon). You would be left with the regression

$$M_x y = M_x  z  \beta_1 + residuals$$

where $\beta_1$ is numerically the same as in the regression above. But now you are able to estimate just one (beta-)parameter with 5 observations compared to three parameters before. So, obviously, the estimates improve, don't they?

Where is our mistake?

We know that $M_X\mathbf{u}= \mathbf{\hat u}$, where $\mathbf{y}=\mathbf{x\beta}+\mathbf{u}$. So even if the error terms are independent, the residuals will not be independent of the other residuals. This is makes the usual assumption of no autocorrelation invalid.

Of course, if we use a HAC estimator then we can circumvent the existence of autocorrelation. However, the HAC estimators that I know of are just of asymptotic behaviour. So, it wouldn't be useful for a small sample size.


Is there an economic term that means the same thing as "personal inflation" I don't have anything even close to what would be considered formal education in terms of economics, so the term "personal inflation," is just something I completely made up. What I mean by personal inflation is that, in terms of living condition, 1000$$ given to someone who is already a millionaire isn't going to be worth very much because, compared to the amount of money that person already had before, 1000$ isn't very much money. However, if that same 1000$ were to be given to someone who had little to no money, the money would be valued at much more and raise the living condition, for at least a little while, of that person much more than it would the millionaire.

What my questions is regarding is what the better term for this "definition" that is explained above is, and/or whether there are terms the fit this description above fairly well, but are slightly different.

What you describe is called "decreasing marginal utility of income". Marginal utility is just an economist's way of saying the benefit you got out of the last unit of something, and decreasing because we think this might be lower at higher levels of income.


Are expensive cities economically threatened by an inability to house low-income workers? Cities like New York, San Francisco, and others enjoy booming local economies that support high-paying jobs in industries like finance and technology. This and other factors have driven the price tenants are willing to pay for rent to astronomical levels, pricing out lower-income workers who then have to make long commutes or take public transit to get to work.

Where I live, the average rent is $2,000, and both that figure and the area affected by inflated rent prices expands every year. As this progresses, help wanted signs and aggressive recruiting for low income jobs have become far more commonplace, indicating a labor shortage in these industries that is steadily increasing as the difficulty of commuting from areas with reasonably priced rent increases.

My city is by no means in a dire labor shortage situation, but I thought it might be interesting to consider a hypothetical city where the area of price-inflated rent becomes so large that basic upkeep from low-paying jobs that "make the world go 'round" is no longer feasible. Is this a real threat to the success of these cities? That is to say, is a labor market correction likely to drive wages up enough to sustain a workforce of extremely long-distance commuters?

Bonus:

Assuming this is a real threat, are rent regulation (San Francisco), high-quality public transit (New York), or anything else effective long term solutions?

Something missing from more traditional analyses is that a significant number of people may move to a new (expensive) location, take a job that pays a wage insufficient to cover living expenses, and burn through their savings until they're forced to relocate.  This happens quite often in "fashionable" locations such as San Francisco.

This is actually a strong argument in favor of a higher minimum wage in such places, even taking a "hard-line" position that such laws only result in the loss of low-paying jobs — because such jobs are a "trap" that lures people (with savings) into an unsustainable situation in which they are exploited until they run out of savings, at which point they are replaced by somebody else with savings to exploit.  Eliminating all jobs that pay wages insufficient to actually live (on an ongoing basis) in expensive markets guarantees that such "exploit the newcomers" economies cannot persist for very long.


Is the set of optimal strategies convex in a single-agent decision choice problem? EDITED with insights from the comment below.

Consider a decision maker who has to choose an action among $\mathcal{Y}\equiv \{1,2,...,L\}$. The payoff from choosing action  $y\in \mathcal{Y}$ depends on the state of the world, $V$, with support $\mathcal{V}$. Specifically, choosing action $y\in \mathcal{Y}$ leads the payoff $u(y,v)$, where $u:\mathcal{Y}\times \mathcal{V}\rightarrow \mathbb{R}$.

Suppose the decision maker has complete information about the realisation of $V$ drawn by nature.

A (mixed) strategy of this choice problem is a probability kernel, $\mathcal{P}_{Y|V}\equiv \{P_{Y}(\cdot| v)\in \Delta(\mathcal{Y}): v\in \mathcal{V}\}$,  collecting probability distributions of  $Y$ conditional on every realisations $v$ of $V$.

Hence, $\mathcal{P}_{Y|V}$ is an optimal strategy of the choice problem  above if  $\forall v\in \mathcal{V}$ such that $P_{Y}(y|v)>0$, and  $\forall \tilde{y}\neq y$
$$
\begin{aligned}
u(y, v)  \geq u(\tilde{y},v). \\
\end{aligned}
$$

Let $\mathcal{Q}^*$ be the collection of all optimal strategies of the choice problem  above, that is
$$
\mathcal{Q}^*\equiv \Big\{\mathcal{P}_{Y|V}: \forall v\in \mathcal{V}, \forall y \in \mathcal{Y}\\
\hspace{6cm}\underbrace{P_{Y}(y|v)>0 \Rightarrow u(y, v)  \geq u(\tilde{y},v)\text{ } \forall \tilde{y}\neq y}_{\text{This is not a linear constraint because of the form "IF ... THEN ..."}}\Big\}
$$

Question 1) The definition of $\mathcal{Q}^*$ just given seems to highlight that $\mathcal{Q}^*$ is not a convex set. This is because it is defined by a constraint of the type "IF ... THEN ...", which is not linear.

Is this comment correct?

Question 2) Consider a payoff function $u(1,v)=u(L,v)>u(y,v)$ $\forall y \neq 1,L$ and $\forall v \in \mathcal{V}$. Consider the following strategies
$$
1) \mathcal{P}_{Y|V}\text{ s.t. } P_{Y}(1|v)=1 \text{  and }P_{Y}(y|v)=0 \text{ }\forall y\neq 1, \forall v \in \mathcal{V}
$$
$$
2) \tilde{\mathcal{P}}_{Y|V}\text{ s.t. } \tilde{P}_{Y}(L|v)=1 \text{  and }\tilde{P}_{Y}(y|v)=0 \text{ }\forall y\neq L, \forall v \in \mathcal{V}
$$
$$
3) \mathcal{P}^*_{Y|V;\alpha}\text{ s.t. } P^*_{Y}(1|v;\alpha)=\alpha P_Y(1|v) \text{, } P^*_{Y}(L|v;\alpha)=(1-\alpha) \tilde{P}_Y(L|v) \text{,  and }P^*_{Y}(y|v;\alpha)=0 \text{ }\forall y\neq 1,L, \forall v \in \mathcal{V}, \forall \alpha \in (0,1)
$$
I believe that the set
$$
\mathcal{B}\equiv \{\mathcal{P}_{Y|V}, \tilde{\mathcal{P}}_{Y|V}, \mathcal{P}^*_{Y|V;\alpha} \text{ }\forall \alpha\in (0,1)\}
$$
is convex. Indeed it seems to me that $\mathcal{B}$ is the convex hull of $\{\mathcal{P}_{Y|V}, \tilde{\mathcal{P}}_{Y|V}\}$.

Correct?

What is the relation between $\mathcal{Q}^*$ and $\mathcal{B}$?

I think that $\mathcal{B}\subseteq \mathcal{Q}^*$. This is because for each element in $\mathcal{B}$, the "IF ... THEN ..." condition defining $ \mathcal{Q}^*$ is satisfied.

Does  $\mathcal{Q}^*\subseteq \mathcal{B}$ too? If my assertion is question 1) is correct, then it should be $\mathcal{Q}^*\supset \mathcal{B}$ because otherwise $\mathcal{Q}^*$ would be convex. But here I'm lost: which element of $\mathcal{Q}^*$ does not belong to $\mathcal{B}$?

The set $Q^*$ is the set of probability distributions over the maximizers of $u$ for each value of $v$. So for a fixed $v$ all the values of $Y$ that have positive probability must give the same utility. That is, if $P_Y(y|v)>0$ and $P_Y(y'|v)>0$ then $u(y,v)=u(y',v)\geq u(\tilde y,v)$ for all $\tilde y\neq y, y'$. Therefore, convex combinations of elements in $Q^*$ must also be collections of probability distributions $P_{Y|V}$ supported over the same elements in $Y$. That is, the convex combination will also be supported over maximizers of $u$ for each $v$.

In conclusion, even though the set of maximizers is not guaranteed to be convex, the set $Q^*$, of distributions, is a convex set.


Lack of skilled IT workforce, but they do not raise wages In my country, Portugal, many IT companies are always complaining in media regarding the lack of skilled IT workforce, but they do not raise wages. We have statistical data from the government that proves what I am saying.

Companies are paying less in 2016 than in 2011 for IT workforce (-10.2%). GEP/MTSSS
From 2014 to 2017 there is a decrease of IT workforce from 110.900 to 104.300 employees. Eurostat - Employed ICT specialists

So, the facts:

Companies are paying lower salaries
The number of it specialists also decrease (probably emigration).

If there is a shortage of workers, as the companies saying, why the law of supply and demand does not apply in this specific labor market?

The factual observations you've listed fit neatly with the law of supply.  You've seen that wages (price) have fallen, and during the same period, supply has gone down.

The fact that companies publicly complain that there aren't lots of cheap, highly skilled workers available is not the same as companies being willing to convert that into effective demand by raising wages.

One common reason companies moan about things publicly, for instance, is that they want the government to intervene to help them.  In this case, perhaps they want government to provide more subsidised IT training, or permit more immigration in the sector, either of which could lower wages further or prevent them from rising.


Replicate average risk-free rate in Wachter 2005, "Solving models with external habit" I might be making a really simple mistake somewhere, but I thought I'd ask anyway. I'm trying to replicate the results in Wachter 2005, "Solving models with external habit". (You can also find the prepublication version here for free.) However, I'm getting a slightly different result for some quantities and I'm wondering if I'm interpreting the parameters correctly. In particular, when I input the parameters listed in Table 1, I get a different average risk-free rate in both the "CC value" case and the "Wachter value" case.

From Tables 2 and 3, we see that the "CC value" case should give an average risk-free rate, $E[r^f]$, of $0.94\%$ and the
"Wachter value" case should give $1.47\%$.
The value of the average risk-free rate can be derived analytically, and is given in equation (11):
$$
E[r^f_t] = - \log \delta + \gamma g - \frac{\gamma (1-\phi) - b}{2}. \tag{11a}
$$
When I substitute in the parameter values from Table 1, I get
$0.47\%$ for the CC parameterization and
$2.47\%$ for the Wachter parameterization.
I wonder if I am annualizing correctly? I'm somewhat confident that I'm inputting the parameters correctly, because the paper does show that for the Campbell Cochrane calibration (CC values), a derived parameter called $S_{\text{max}}$ should be equal to $0.0939$,
where
\begin{align}
\bar S &= \sigma_\nu \sqrt{\frac{\gamma}{1-\phi - b/\gamma}} \tag{5}\\
s_{\text{max}} &= \bar s + \frac 12 (1 - \bar S^2), \tag{6}
\end{align}
Lower case letters denote the natural log of the capitalized variable. When I substitute in the "CC values", I get $S_{\text{max}} = 0.09384$. So, this seems correct. The paper implicitely reports this parameter in a footnote:

Any thoughts as to why I'm getting different values for the average risk-free rate?
I've included Python code below to show what I'm doing.
# Campbell Cochrane parameters
ann_fact = 12 #annualization factor. Monthly
g = 1.89 / ann_fact / 100
sigma_v = 1.50 / np.sqrt(ann_fact) /100
gamma = 2.00
b = 0.00
phi = (0.87)**(1/ann_fact)
delta = (0.90)**(1/ann_fact)

# ## Wachter 2005 Parameters
# ann_fact = 4 #annualization factor. Quarterly
# parameterization_name = 'Wachter (2005)'
# ann_fact = ann_fact
# g = 2.20 / ann_fact / 100
# sigma_v = 0.86 / np.sqrt(ann_fact) / 100
# gamma = 2.00
# b = 0.011
# phi = (0.89)**(1/ann_fact)
# delta = (0.93)**(1/ann_fact)

bar_S = sigma_v * np.sqrt(gamma/(1 - phi - b/gamma))
bar_s = np.log(bar_S)
s_max = bar_s + 1/2 * (1 - bar_S**2)
Smax = np.exp(s_max)
print("Smax ", Smax) # CC1999 should be 0.0939
r_f = (-np.log(delta) + gamma * g - 1/2 * (gamma * (1-phi) - b))
print("Average risk-free rate", r_f * ann_fact * 100)
# CC1999 should be 0.94 %
# Wachter 2005 should be 1.47 %

I think the answer just comes down to rounding error. The reported values of $\delta$ are annualized. In the case of the CC1999 calibration, the annualized $\delta$ is $0.90$ and the implied monthly is $\delta = 0.991258$. Because of the compounding over 12 periods (4 in the Wachter 2005/2006 case), the average risk-free rate is very sensitive to small changes in this value. Taking the other parameters as given and solving (11a) for the appropriate $\delta$ in each calibration leads to the following annualized values that are needed to reproduce the desired risk-free rates:
\begin{align}
\delta &= 0.8957829809431154 \text{ (instead of 0.90 in CC calibration)} \\
\delta &= 0.9384246954559854 \text{ (instead of 0.93 in Wachter calibration)}
\end{align}
Below is the code to verify this.
# Campbell Cochrane parameters
ann_fact = 12 #annualization factor. Monthly
g = 1.89 / ann_fact / 100
sigma_v = 1.50 / np.sqrt(ann_fact) /100
gamma = 2.00
b = 0.00
phi = (0.87)**(1/ann_fact)
# delta = (0.90)**(1/ann_fact)
delta_CC = (np.exp(- 0.94/ann_fact/100 + (gamma * g - 1/2 * (gamma * (1-phi) - b)) ))**ann_fact
delta = (delta_CC)**(1/ann_fact)

# ## Wachter 2005 Parameters
# ann_fact = 4 #annualization factor. Quarterly
# parameterization_name = 'Wachter (2005)'
# ann_fact = ann_fact
# g = 2.20 / ann_fact / 100
# sigma_v = 0.86 / np.sqrt(ann_fact) / 100
# gamma = 2.00
# b = 0.011
# phi = (0.89)**(1/ann_fact)
# # delta = (0.93)**(1/ann_fact)
# delta_wachter = (np.exp(- 1.47/ann_fact/100 + (gamma * g - 1/2 * (gamma * (1-phi) - b)) ))**ann_fact
# delta = (delta_wachter)**(1/ann_fact)

bar_S = sigma_v * np.sqrt(gamma/(1 - phi - b/gamma))
bar_s = np.log(bar_S)
s_max = bar_s + 1/2 * (1 - bar_S**2)
Smax = np.exp(s_max)
print("Smax ", Smax) # CC1999 should be 0.0939
r_f = (-np.log(delta) + gamma * g - 1/2 * (gamma * (1-phi) - b))
print("Average risk-free rate", r_f * ann_fact * 100)
# CC1999 should be 0.94 %
# Wachter 2005 should be 1.47 %
```


tariffs and inflation tariffs lead to inflation; is this the same as inflation from supply constraints or demand drive, given that the tariff is removed from the money supply? How should the fed be thinking about this in the context of raising interest rates if they use inflation a gauge of how much to hike?

Across the board tariffs (ex: sales tax) can result in higher prices or "inflated" prices. This is usually not what economists refer to when they say inflation. Inflation that the fed is concerned with, is an across the board increase in all goods as a result from changes in the money supply. Something, that the fed can directly influence, whereas the fed has no control over taxes.

Fiscal policy (Government) and monetary policy (The Fed) can sometimes coordinate actions for a desired result.

For instance: Following the 2007/8 financial crisis, the united states paired subsidies and incentives ,effectively lowering sales tax (fiscal policy), with low interest rates (monetary policy) to stimulate the economy.

You can imagine that a coordinated policy concerned with high inflation rates might do the opposite.


pdynmc package in R | GMM issue I am a student, currently working on my Bachelor's Thesis. And as the next deadline approaches, I hope someone could give me a hand on my constant issue:
I need to estimate the next model with GMM estimator:

So far I was using pgmm function in plm package. You can check other topics of mine with the issues I have faced, and I still did not solve them.
Hence, I tried to move to another package pdynmc authorized by Dr. Markus Fritsch.
Here is a part of my dataset filtered with firms having at least 5 years of observations: https://drive.google.com/file/d/1dFUgXp8e1K0CdqMTpbtfviDq9vgJNOxW/view?usp=sharing
I am not sure you can get a context of it, but I need to estimate similar to this model as:
A change or growth in the dependent variable from t to t+1 (ΔTFPt+1) = GMM instruments, as far as I understood, of the, lagged dependent variable (so lagged change in the dependent variable) (ΔTFPt, and ΔTFPt-1) + change in one other variable (ΔDebt), the measure of needed variable (financial friction) + interaction between last 2 variables (financial friction * ΔDebt) + control variables (firm age, firm size or log(TA), change in sales or ΔSales).
In my dataset, needed variables are:

ID - is an ID of the companies
Year - the year of observation
ta - log of total assets
ff1-5 - financial friction variable of the companies
LVomegaACF_A - is a TFP measure
domegaACF_A - is ΔTFP measure
ddebt -  ΔDebt variable
dsales -  ΔSales variable
Age - age of the companies

I tried to use the package as follows:
model <- pdynmc(dat = LVcheck, varname.i = "ID", varname.t = "Year",
                varname.y = "domegaACF_A", lagTerms.y = 2,
                fur.con = FALSE, varname.reg.fur = c("ddebt", "ff1", "Age", "ta", "dsales"),
                lagTerms.reg.fur = c(1,1,1,1,1),
                fur.con.diff = TRUE, fur.con.lev = FALSE,
                include.dum = TRUE, dum.diff = TRUE, dum.lev = FALSE, varname.dum = "Year",
                w.mat = "iid.err", std.err = "corrected", estimation = "onestep", opt.meth = "none",
                use.mc.nonlin = FALSE, use.mc.diff = TRUE, use.mc.lev = FALSE, include.y = TRUE)

And the error I receive:
Error in pdynmc(dat = LVData_A, varname.i = "ID", varname.t = "Year",  :
  Insufficient number of time periods to derive linear moment conditions.

The question is the following, why the number of periods is insufficient if the only firms are displayed that have at least 5 years of observations?
And also, is my code written accordingly to my model specification, and all the arguments needed are written correctly.
I would really much appreciate your responses and any support from your side. Here is the package description https://rdrr.io/cran/pdynmc/man/pdynmc.html
Despite the fact, everything is quite detailed there, I still do not understand what is wrong with what I am doing.
Thank you very much in advance.

I apologize for my previous comment; it doesn't make much sense.
Upon inspecting your data, I've found that there are IDs for which you only have observations for two years. The problematic IDs are: 9231, 24873, 24907, 78341, 78998, 90246, 92673, 92787, and 93846. You can see the full frequency table here.
By looking at the source code for the package, we can see what causes that error message to appear:
  if((use.mc.diff | use.mc.lev) && (length(unique(dat[, varname.t])) <
 3)){

stop("Insufficient number of time periods to derive linear
     moment conditions.")
}

which I understand as: if the length of the vector containing the unique dates is less than 3, then the estimation cannot be computed.
Looking at the package help, we can run the example code provided:
## Load data from plm package

    if(!requireNamespace("plm", quietly = TRUE)){

     stop("Dataset from package \"plm\" needed for this example.
         Please install the package.", call. = FALSE)
    }

    else{

    data(EmplUK,
         package = "plm")  dat <- EmplUK  dat[,c(4:7)] <- log(dat[,c(4:7)])

         ## Arellano and Bond (1991) estimation in Table 4, column (a1)

     m1 <- pdynmc(dat = dat, varname.i = "firm", varname.t = "year",
                  use.mc.diff = TRUE, use.mc.lev = FALSE, use.mc.nonlin = FALSE,
                  include.y = TRUE, varname.y = "emp", lagTerms.y = 2,
                  fur.con = TRUE, fur.con.diff = TRUE, fur.con.lev = FALSE,
                  varname.reg.fur = c("wage", "capital", "output"), lagTerms.reg.fur = c(1,2,2),
                  include.dum = TRUE, dum.diff = TRUE, dum.lev = FALSE, varname.dum = "year",
                  w.mat = "iid.err", std.err = "corrected", estimation = "onestep",
                  opt.meth = "none")
  summary(m1)
    }

which works just fine. Furthermore, if we run the following code:
length(unique(dat[, "year"]))

we get 9 as the output, which is clearly greater than 3. However, if we do the same using the data you have provided:
length(unique(datacheck[, "Year"]))

the result is 1, which is smaller than 3, just as the error message pointed out.
I can't figure out why that is happening. My guess is that perhaps you need at least 3 unique sequences of years or something else, because you do have enough observations with more than 3 years available.
While this is not a definitive answer, it's too long to post as a comment. Hope it helps anyway.
EDIT:
Source code
Vignette


Why can we finance government deficit by having a trade deficit? I encountered an example question regarding government deficit as follows:

$G-T=(S-I)-(X-M)$

where $G$ is government spending, $T$ is net tax (tax minus transfer payment), $S$ is total private sector saving, $I$ is investment, $X$ is export and $M$ is import.

Suppose deficit $G-T=\$84249$, $S-I=\$58558$, so $X-M=-25661$.

The example question states that: to finance the deficit, private saving must exceed investment by $\$58558$, and to finance the rest of the government deficit, foreign imports ($M$) must exceed exports ($X$) by $\$25661$.

My question is, if there is a trade deficit (import is greater than export), that means the government buys imported products more than it sells the exported products; the government spends more than it earns, then how can it finance the government deficit?

This really confuses me. Thanks for any help!

To answer the question, I feel it is needed to correct one misunderstanding first: trade deficit does not mean the government buys imported products more than it sells the exported products. Rather, it means the imported products are more than exported products, which does not indicate whether the products are consumed or invested by the government or the private sector.

And this question relates the double deficit problem (also called twin deficit).
Let's start from the no tax situation, and there is:
$$Y-C-G=I+NX$$
Since the LHS represents the total saving of the country (denoted as $S_{T}$), we have:
$$S_{T}-I=NX$$
Recall that saving is the surplus of a country's production that is not consumed by private and government.That's why total saving equals its investment ($S_{T}=I$) if it is a closed economy ($NX=0$).

In the perspective of an open economy, the consumption of a country's production is not only domestic spending on domestic products from the private sector ($C_{d}$) plus domestic spending on domestic products from government ($G_{d}$), but also foreign spending on domestic products ($X$). The private and government spending on foreign goods, as well as investment on foreign goods and services, should be eliminated from GDP or $Y$, and we have $M=C_{f}+G_{f}+I_{f}$ ($Y=C+G+I+X-M$)

Come back to the equation $S_{T}-I=NX$. The RHS is the trade balance, and the LHS can be view as the balance of this country. If the LHS is negative, it means the country has less saving than it has invested, which means it is the borrower in the global capital market (there is a capital inflow) in order to purchase foreign goods and services. This is caused (or accompanied) by a trade deficit, because the country produces ($X$) less than it consumes ($M$), which requires it to borrow money in the global market (if not, say, in the case of no capital flow, its currency will depreciate to balance $NX$, due to an excessive supply of domestic currency caused by trade deficit).

Adding tax is simply separate total saving into private saving and tax, which is just slightly different from the original equation, that is,
$$S+(T-G)-I=NX$$
As government spending is always exogenous, and others factors are relatively stable, the double deficit hypothesis supposes that an increase in government deficit will result in an increase in trade deficit, accompanied by an increase in capital inflow to finance the extra government spending.


Is market failure the same allocative inefficiency? From an IB Economics book (in the microeconomics section):

The part in blue suggests market failure is the same as allocative inefficiency, but the part in red seems to suggest that market failure is one cause or type of allocative inefficiency.

What is the difference between the two, if any?

In this context 'market' usually means some allocation mechanism. For example free market, or a market where someone has monopoly power. Allocation of goods and resources can be done in other ways: e.g. you can divide all resources evenly among production processes and all goods evenly between consumers. This will generally result in allocative inefficiency (were all production processes equally important and efficient?). But it is not market failure because there was no market, just an allocator/social planner.


"$\backslash$" notation question in definition for Nash equilibrium A definition in my book states

  $ \sigma^* \in \Delta$ is a Nash equilibrium for $\Gamma$ if for all $i \in I$ and all $\mu_i \in \Delta_i, \; U_i (\sigma^*) \geq U_i(\sigma^* \backslash \mu_i)$.  The set of Nash equilibria for $\Gamma$ is denoted $Eq(\Gamma)$

What does the $\backslash$ here represent, divide or element not included?

Informally, a (possibly mixed) set of strategy choices is a Nash equilibrium if no player can expect to benefit by changing strategies while the other players keep their strategies unchanged.

Here I would interpret your definition as $\Delta$ being the set of all possible strategy choices across all players, $\sigma^*$ being a set of strategy choices which will be a Nash equilibrium if it satisfies the conditions, $i$ being a player, and $\sigma^* \backslash \mu_i$ being a change in player $i$'s strategy while the other players' strategies stay the same.  So I would take $\backslash$ having a type of "not included" interpretation.


Phase Diagram - growth model The dynamics of Ramsey-Cass-Koopman growth model is usually summarized in phase diagrams with the 2 equations (conventional symbols apply):
\begin{align*}
\frac{\dot c}{c}=&\frac{r-\rho}{\theta}\\
\\
\dot k =& f(k) - (n+g+\delta)k
\end{align*}

From these equations you work out the equilibrium and whether it is a saddle path.

However this analysis is static; some of those parameters are time varying (e.g., 1950 population growth isn't the same as in 2017). How can you represent these changes in this essentially static graphical framework?

In control systems engineering, there was the "Aizerman Conjecture" (the transliteration of Aizerman varies) that argued that a linear time-varying system was stable if the parameters (state matrices) at all times corresponded to stable systems. However, counter-examples were found to this conjecture.

About the only way to show that a time-varying system is stable (for example) is to find a Lyapunov function that holds at all times. This is generally going to be difficult.

Something like graphical analysis would be technically invalid, as you are violating assumptions that are in place. You are pretty much going to have to derive any results from first principles.


Does the amount of inward investments affect inflation? I understand that inflation has an impact on the foreign direct investment. But can the opposite be true? The amount of foreign direct investment has an impact on the inflation of the country that receives the investment?

Thanks

Yes, FDI and Inflation are correlated.

Think of it this way, if foreigners buy assets, that reduces the supply available for domestic consumption, driving up prices, and if they buy them from citizens, it increases the domestic supply of money.

That being said, the relationship can be complicated because if there's substantial trade of real assets between foreigners, the effect might not trickle down to the domestic populace.

Ex 1:

  Alice and Bob are citizens of Country X. Charles is a citizen of Country Y. At equilibrium, sans FDI, Alice and Bob price the only pineapple at \$3. If Charles buys that pineapple for \$4 from either of them (assuming symmetric preferences for A & B), he has injected \$1 into the economy of Country X and removed 1 pineapple, consequently the supply of money has increased relative to the supply of assets, and prices should increase.

Ex 2:

  If Charles had left the pineapple in Country X, and David, a citizen of Country Z then bought it from him for \$5, and Charles did not invest any of that profit into Country X, the supply of capital in X would not be increased by this transaction, nor would the supply of assets, consequently the ratio of money to goods hasn't changed, and country-wide price levels shouldn't fluctuate.

Ex 3:

  Assuming there was a second pineapple that Alice or Bob held, they could then borrow against it at a higher value (given the new equilibrium price and FMV), leaving more money in the economy and consequently increasing the price of sponges for Bob as the ratio of assets to money would have changed in situ.

Note: This is an off the cuff explanation, not a formal model


What are financial service “passports”? What are “passports” in the context of financial services?

I have also heard the term “passporting rights.”

The EU passporting system for banks and financial services companies enables firms that are authorised in any EU or EEA state to trade freely in any other with minimal additional authorisation. These passports are the foundation of the EU single market for financial services.


How does elasticity impact market efficiency? I am wondering how the price elasticity of demand (or supply) impacts market efficiency. A more specific question - can we say that in a monopolized market elasticity will be low, and that in a competitive market the competition allows the elasticity of demand to increase? Said another way, a lack of competition places a ceiling on the elasticity of demand, when there is good competition, the elasticity is not bound and reaches the correct market equilibrium value.

Can a regulator use low elasticity as evidence there is a lack of competitiveness?

Can an increase to price elasticity of demand increase consumer surplus while reducing producer surplus?

Would be great to hear some layman thoughts,a s well as references to academic literature (I didn't find much on the topic in my search).

Good question

If we examine edge cases of full elasticity, then we get:
supply or demand with zero impact of quantity on price: no matter the quantity, price will remain the same

Looking at elastic demand, that mean that for suppliers the MC function will arrive to the particular demand price. No special change in efficiency

Looking at elastic supply, that means that the consumer surplus will derive from the fixed price

Based on theory, there will be no loss of efficiency (a certain quantity that will be lost) when adding taxes or subsidies

The common real case of full elasticity, is for imported or exported goods, where the world market's equilibrium for the product imported /exported is unaffected by local market
In this case, either consumers get cheaper prices for products, comparing to a similar product manufactured locally
or get extra prices for local goods sold abroad


Is there a cost to running the NewYork Stock Exchange? Does it affect the price of stocks sold on the exchange? Is there a cost to running the New York Stock Exchange? For example electricity costs, management costs,  rents on the building(s).  How do these costs (if any)  affect the price of stocks sold on the exchange?

Yes, there's a cost to running the New York Stock Exchange. There's the premises, utility bills, staff, systems, and so on.

Yes, this affects the price of stocks on the exchange. The exchange recoups its fees, ultimately, from the companies listed on it, pushing up their costs.

However, for pretty much all companies listed on it, the stock exchange fees are a tiny part of their overall costs. Which means that the impact on the stock prices is pretty much negligible.