Are there studies that empirically validate results in contract theory? I have started learning contract theory of late, and I find some of the assumptions and findings of the theory are too technical. Are there empirical studies that have validated concepts in contract theory? Or are there papers that talk about its relevance to practitioners (economists or operations managers)?

An excellent (if slightly old) survey on the empirics of contracts can be found in

  Pierre-André Chiappori and Bernard Salanie (2003) "Testing Contract Theory: A Survey of Some Recent Work", in Advances in Economics and Econometrics, vol 1, M. Dewatripont, L. Hansen and S. Turnovsky eds, Cambridge University Press.

It is mostly non-technical. Bernard Salanie (literally) wrote the book on contract theory. An ungated pdf of the chapter is available here.


Arrows Theorem: Dictatorship How does Arrow's Impossible Theorem show that the only aggregation rule that works is Dictatorship. Couldn't we have a Non-dictatorship with IIA or a Non-Dictatorship with Pareto Efficiency?

Arrow's Impossibility Theorem merely states that (with a ranked voting method) we cannot simultaneously satisfy IIA, Pareto Efficiency, Non-dictatorship and unrestricted domain.  You can satisfy any subset of those conditions, though.  So Non-dictatorship with IIA is fine — so long as you're willing to drop Pareto Efficiency.  Similarly, Non-dictatorship and Pareto Efficiency can be satisfied — if you drop IIA.

You can also use a system that is not ranked, such as Cardinal Voting although there is some debate as to  how to adapt the IIA criterion in such cases, and some such systems nonetheless fail different criteria not addressed by Arrow, such as the Majority Criterion.

(Unrestricted domain has more to do with how to decide which candidates are eligible to appear on the ballot, and which voters are allowed to vote in general, and so isn't usually considered as one of the criteria that can be relaxed... it's more a formality in Arrow's theorem, than the other criteria.)


Beliefs about equilibrium paths with overlapping strategies This is a basic question, but I cannot find the answer to it. In short, equilibria are often conditioned on a specific strategy. E.g., given A's strategy, B updates her beliefs to xyz. My problem is that often an action could be a part of the equilibrium path for many different strategies, so how does B know which strategy A is following?

Below is an example that might clarify the question.

Consider a classic signaling game (2 types, 2 actions) à la Spence. Suppose furthermore that there are two equilibria: one pooling, one separating. In the pooling eq., say, both types of senders send "Low". In the separating, the "Strong" type sends High, the Weak type sends Low.

The part that I struggle with is: how does the receiver know what equilibrium path they are in? For example, say the receiver observes "Low". If we are in "pooling world", then the receiver's posteriors are just going to be her priors. If, however, we are in "separating world", then she can update to p(Weak|Low)=1. But observing "Low" alone does not tell the receiver what strategy the sender is following, so how can she possibly update her beliefs? It seems to me that she would need to have beliefs not only about types, but also about strategies being followed.

Sorry if this is idiotic, but this has puzzled me for a while.

In signaling games, beliefs as well as strategies constitute an equilibrium. In other words, beliefs are a part of the equilibrium object. By deciding which equilibrium to focus on, you also (partially) determines the beliefs consistent with that equilibrium. The key restriction is that beliefs be derived from Bayes' rule using sender's strategy and the prior.

Consider the following game, where P1 is the sender and the common prior is Type I realizes with probability $p$.

Separating equilibrium

In a separating equilibrium, different types choose different actions (Type I chooses $A$ and Type II chooses $B$). Then, upon seeing action $A$, P2 (the receiver) updates her belief using Bayes' rule:
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(0)(1-p)}=1
$$
Therefore P2's conditional belief at node $x$ is $1$. Similarly for her belief at the information set $\{x',y'\}$

Pooling equilibrium

Now, consider a pooling equilibrium, where both types choose $A$. Then, upon seeing action $A$, P2's belief is calculated as
$$
\mu(\text{Type I}|A)=\frac{\Pr(A|\text{Type I})p}{\Pr(A|\text{Type I})p+\Pr(A|\text{Type II})(1-p)}=\frac{(1)p}{(1)p+(1)(1-p)}=p
$$
which is the same as her prior. This makes sense because P1's action provides no additional information in a pooling equilibrium. (P2's off-equilibrium belief can be arbitrary, unless you want to apply refinements such as consistency of the sequential equilibrium).

The bottom line is: you cannot talk about an equilibrium separately from the beliefs. By talking about a particular equilibrium, you also implicitly restrict the set of beliefs consistent with the said equilibrium.


What is the main cause of differences in the costs of living? Following question is not formulated in technical language of economics, since I am not an economist and I am mostly interested in accessible answers. Yet, I think it goes deeply into theory of economy and I hope it is not off-topic.

In different regions of the world, costs of living are very different. For simplicity, let us consider two countries which use the same currency (for example euro). In country A, there are high salaries, high housing costs, high costs of services, while in country B, all these quantities are lower. If we compare how much haircuts or meals in restaurant can citizen of country A buy for his salary, the number might be similar to what citizen of country B can buy for his salary. The prices of commodities often do not differ that much, but they tend to differ as well. The quality of services is similar. My question is: What is in the core of the difference in costs of living? What causes them?

To clarify, I am not interested in answers like "country A is more developed", or "country A is more rich", since it doesn't explain the issue, it just gives it another name. I would like to understand the mechanism. I have two hypotheses, which might be the cause, but I am not able to judge which one is more true, or if they are correct at all.

Hypothesis 1: History. During creation of the common currency, there was some agreement how much the national currencies are worth. (Based on international trade and older history.) This set up the level of costs of living to some value. If there is not much trade and migration between country A and B, costs of living equilibrate only slowly.

Hypothesis 2: Transfer from other sectors. If there is some high technology and well paid industry in country A, employees of this industry can afford better services and can afford to offer more money for housing, which increases salaries of all people in services and possibly transfers to all sectors of society.

The United States has had a common currency for over two hundred years.  The cost of living in rural Georgia has not equalized with that in New York City during that time.  If anything, they've diverged.

Land in New York City is more expensive than land in rural Georgia.  As a result, rent is more expensive in New York City.  This has a cascading effect.  Because residential rent is more expensive, wages need to be higher.  Because commercial rent and wages are higher, prices of products and services need to be higher.  Which makes wages need to be higher again.

Note that commodities will often have very similar prices even in these conditions.  This is especially true of things that can be ordered from outside the region.  Even something like a haircut might be similar just because of economies of scale.  In New York City, there are more people needing haircuts, so a moderate price may still produce more revenue.  And of course, there are places that charge much more for a haircut.

Theoretical differences can be caused by something as simple as preferences.  Or by history.  Or by a single industry.  Or by some combination of those.  For example, New York City has historically been a financial center and had the kind of night life that a large city can.  That's history, preference for an advanced night life, and a special industry.  The reasons in your hypothetical countries are also likely to be complicated and probably not entirely understood.

In the end, the most we can say is that people are willing to pay more to live in New York City than rural Georgia.  If you went around and asked people in New York City, they might give a bunch of different reasons.  One person has always lived there.  Another came to work on Wall Street.  Another tried several places and just prefers it.  Someone else came to be on Broadway.  Unfortunately, it's unlikely that we can point to a single economic reason unless we pick something bland like, "because they could."

Given two actual countries, we might be able to do a little better.  That's why I've been picking on the New York City/rural Georgia comparison.  That at least gives something to discuss.  The big point that I wanted to make is that your assumption that historical differences would disappear is incorrect.  Historical differences could persist.  Prices don't have to equalize in a common currency area.


Creating a pragmatic, useable (not theory heavy) proxy for a currency basket for a set of economies This is currently, a gedanken experiment, but may well form the basis of a whitepaper - or better still, an actual project.
I am trying to come up with a practical way of creating a "currency index/basket" (of sorts) between a set of countries who want to form some kind of (geo?)political alliance, but currently, have their own currencies.
The idea is that if a freely convertible currency exists (ignore who "controls" the currency - for the purposes of this thought experiment), then the countries in the "allowed set" can freely interchange between each others currencies - without having to go through the Dollar, EUR, Reminbi or other external currency first.
The thinking is very much along the lines of the Euro (although the constituent countries need not be restricted to the same geographic zone in my case).
So, here are the salient points:

A set of independent sovereign states exist, and they would like to increase trading amongst themselves
Currently intra-set trading involves triangulation involving one of the majors.
Wish to create an index that is effectively a weighted index of the currencies within the aforementioned set
The index should be pragmatic, and ideally take into account inputs that affect people on the ground (i.e. cost of a basket of household items, factory inputs, market price of any natural resource that forms a portion of GDP revenue for the country  etc. - rather than things like M1,M2 and M3)
The index should be something that can be calculated on a daily basis - so the value can be set at the beginning of the day (akin to London metals price fixing - or indeed the fixing of LIBOR rates at the beginning of the day)

My questions are this:

What would be suitable inputs to capture (on a daily basis)?
How can these inputs be weighted to create the currency index? (i.e. suitable methodology etc).

There is an economic literature on optimal currency areas. However, that is related to the adoption of a single currency.
As described in the question, a basket is just a new asset that is a weighted allocation to underlying currencies.
Once the weightings are fixed, the fair value of the basket is the weighted value of the component currencies, which can be updated in real time.
The only way that this basket changes anything in the real economy is that entities are willing to hold it and trade it.
The issue is straightforward: people and firms want their local currencies, not the basket. What they want is to translate the foreign currency to theirs.
Banks can offer quotes for any currency pair. It is only in the worst case that the quotes would have a bid/ask that is as wide as doing a pair of transactions versus a major currency (often USD, EUR). The only way a basket can improve matters is to have a tighter bid/offer versus the national currencies than they have versus the major currencies. How plausible is that outcome?
The weighting could be done in any number of ways, but the basket has to meet the constraint that people want to own it. The weighting has to reflect the ability to buy short-term financial assets in the underlying currencies. Realistically, that determination has to be done by the entities managing the basket.


Interpretation of Solution Concepts I was wondering whether there is a neat overview over different interpretations of game theoretic solution concepts such as Nash equilibrium, Sequential Equilibrium and the like.

Textbooks I found incredibly lacking in this respect (they have the tough job of getting the mathematical definitions across). Often, for Nash equilibria they allude to some "consistency" or "steady state".

My question is therefore both a reference request and perhaps a call to create a nice overview over different interpretations of equilibrium concepts in game theory. If you feel this question is too broad, it may make sense to narrow it to Nash equilibria.

One reason why there exists no ultimate overview is that these issues are still under debate. A great entry point would be the survey "Foundations of Strategic Equilibrium" by John Hillas and Elon Kohlberg in Volume 3 of the Handbook of Game Theory with Economic Applications. A preprint version of the survey can be found here.


Estimating the rental rate of capital from data Take the classic optimization problem of the neo-classical firm:

\begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{maximize (over $K, L$)} & f(K, L) - RK - WL
 \end{array}
\end{equation}

The first order condition equates the marginal product of capital with the rental rate of capital $R$.

Which raises the question...

How do macroeconomists typically estimate a time series for the rental rate of capital using U.S. data?

(Disclaimer: I don't do macro, but my curiosity has been sparked.)

One approach is to use financial market data to get $R_t$, use another first order condition that the rental rate of capital equals the nominal interest rate plus depreciation (i.e. Hall and Jorgensen). Backing a rental rate out of financial market data is not obvious though! In financial markets, prices vary based upon risk and time.

time dimension: Long-term rates are typically higher than short-term rates. When macro-economists and macro-models talk about the rental rate of capital, what's the time frame?
risk dimension: Eg. callable bonds have high yields than non-callable bonds. Debt and equity may have different expected returns based upon risk.
inflation dimension: A fixed nominal rate is a stochastic real rate depending on realized inflation, and the expected real rate is the nominal rate minus inflation expectations. The real rental rate would add back inflation expectations.

A completely different direction is taken by Casey Mulligan where he sticks entirely with Bureau of Economic Analysis (BEA) data.

I don't follow this literature, and I don't have a sense of the range of approaches that are considered sensible in modern, empirical macro.

Time series on rental price of capital can be estimated using $$r=\frac{P_k}{P}(i-inf+\delta)$$
here, $P_k$ is the price of capital goods (price index for capital goods), $P$ is a deflator, $i$ is nominal interest rate, $inf$ is an inflation rate and $\delta$ is depreciation rate of physical capital stock.


Endogeneity problem with a dummy variable I have a panel data model as following,
$$Y_{it}=intercept +B_1X_{it}+B_2Q_{it}+error_{it}$$

where $i$ is for firm and $t$= time.

$Y$= trade credit demand, $X$= inventory cycle, as inventory moves faster firms demand more trade credit. On the other hand because trade credit means extra inventory, the cycle would be affected by it, too.

I know that the model suffers from endogeneity because $Y$ affects $X$ and in return $X$ affects $Y$. Would it be a valid procedure if I tried to work around this issue by assigning a dummy variable and not use $X$ in the analysis at all by calculating first yearly average for the cycle and assign 1 if a firm is above the average for that year and 0 if below it.  Maybe the code I am using may clarify any vagueness.

in the R code below.

mydata=mydata%>%group_by(year)%>%mutate(mean_x=mean(x))
mydata=mydata%>%mutate(Dummy=case_when(X>=mean_x~1, TRUE~0))

and the model becomes as following:

$$Y_{it}=intercept+B_1 Dummy+B_2 Q_{it}+error_{it}$$

I hope I am clear enough. thanks in advance!

No,

I don't see how the dummy variable you are proposing would give you the same (or even similar) analysis as in the original model.

I would recommend dealing with the endogeneity in another way. IV estimation, or one of the more recent methods, difference-in-difference, matching etc. One of the will be suitable.


How come websites such as Khan Academy and Anatomy Zone do not charge for their services? I've noticed that most educational websites do not charge for their services, even if they are maintained by only one or very few people.

Some of these websites are so comprehensive that they require almost a full time job to maintain.

Why do these websites not charge for their services? (I doubt they make enough money from youtube to make it profitable).

Khan Academy is a 501c non-profit. They do some fundraising, they have some regular backers, and some big time contributors like these.

I imagine it's a similar case for Anatomy Zone. They are affiliated with a bunch of other non-profits that you can see on the bottom of their front page.


What is the difference between VAT and corporate tax Many times I hear that VAT (value added tax) is a "good" tax, while corporate tax is a "bad" tax/ However, both taxes take a cut from the difference between the income of a company and its expenses, and seem therefore to be economically equivalent. Can anybody explain the difference to me?

Both taxes take a cut from the difference between the income of a
  company and its expenses.

Wrong. Companies collect VAT as though they were tax-collectors on behalf of the state. This is why the amount of VAT that a company pays to its suppliers' is subtracted from the VAT that the company has collected from its customers, and only the difference is paid to the State.

As to whether they are "good" or "bad" taxes, and which is which, as usual it is a complicated matter (and when you read/hear such assertions, make your eyes or ears walk away), a matter that has to do not only with businesses but with the macroeconomy as a whole, while it also depends on the current state of the business cycle, whether there is a depression, etc.

ADDENDUM
Responding to the comment by the OP to this answer, here is a schematic view of the matter

V.A.T is not part of the Revenues/Sales of a company. It is a tax imposed on consumers, and collected by the company on behalf of the state.

Imagine a situation without V.A.T., i.e. assume that currently only the first column of the above exists. Then V.A.T. is enacted into tax law. The Profit and Loss statement of the company will not include it, neither in the Revenues side, nor on the Expenses side, it remains unchanged. The amount collected from customers goes up, but the Revenues of a company don't. And if among the company's customers are other companies, they will not see VAT as an increase in cost, because they will get to offset what they pay with what they collect from their customers. In other words, V.A.T is a tax imposed on final private consumers, and everybody else is involved as a tax-collecting entity only. Note that V.A.T is charged and collected irrespective of whether a company has profits or losses.

In a very general sense, all taxes are "equivalent" in the sense that they have the same goal: bring part of output under the ownership and control of the state. But it matters greatly how do they affect the "before-the-State" economy. In this particular case, V.A.T makes products more expensive for final consumers, and it is about how they will react to these increased prices. On the other hand, Income tax cuts directly from a company's profits, if profits exist, and so it affects the incentives of businesses to go into business.


Why do people buy negative interest rate bonds? I get that people buy long term government bonds, because of how safe it is, and if there would be a recession the government would just print more money and could pay it back.

But why would anyone buy a bond, that yields a negative return? If i would just keep my money in fiat, it would just depreciate in the 10 years, but after the 10 years it would still be more than investing it in a safe negative rate.

Some pictures and text are from Schroders, Marketwatch and other websites, but I don't remember all them.
Of note, Germany first sold negative-yielding bonds in Aug 2019, €2bn worth of 30-year bonds that offer no interest payments at all.
1. Better than holding cash
Cash is obviously most liquid, but some central banks like Japan impose negative interest rates on cash deposits, which trickles to the rate banks charge institutional investors.
Then highly liquid but only slightly negative yielding government bond looks better! Because the UK government unlikely will default, bond buyers are just paying a small sum to the government to guard your money, like in a vault!
2. Wagering that there are other “bagholders.”
Investors who buy negative-yielding bonds are betting on the value of the securities to keep rising. Although investors buying bonds with subzero interest rates are paying for the privilege to hold a bond, they can profit way above their cost if the security’s price rises.
For example, if you expect a central bank to buy more assets,  bond buyers can rely on the central bank to hoover up their negative-yielding securities.
In July 2020, an auction for €4 billion euros of 10-year German government bonds TMUBMUSD10Y, 0.644% sold at $-0.26%$, but at a premium price of 102.6 cents to the euro. The benchmark bund is now trading at a price of a 106.9 cents to the euro,. So  investors who bought this at auction would've gained around 4% from the price increase alone.
3. Bonds as deflation hedge
When prices fall, most asset classes don't perform well in deflation. An exception are fixed interest government bonds! Because they fixed  their coupon and principal, they retain their value and will positive real (inflation-adjusted) return, if inflation $<$ their yield. In other words, a negative yielding bond can  positive real return if there is deflation. So you can hedge against deflation with negative-yielding bonds.
4. Currency hedging can transform negative yields into positive yields.
Negative yields don’t mean negative income for some bond buyers. Unlike European and Japanese investors, US investors are often paid to hedge against fluctuations of foreign currencies because U.S. interest rates are much higher than in other developed markets like Europe and Japan.  Currency hedging can annualized return 3% for U.S. investors, like  American fund managers, buying negative-yielding  euro-denominated debt like European government bonds, according to Jens Vanbrabant, senior portfolio manager at Wells Fargo Asset Management.
In Sep 2009, the 10-year German Bund yield at -0.6% looked unattractive to US investors.But currency hedging is priced at the difference between US and German short term interest rates, which can be positive even if a central bank cuts interest rates. Since US short term interest rates are much higher than Germany's, US investors are paid money to hedge euro exposure! With currency hedging, US investors can earn 2.2% when investing in German 10 year bonds (see Figure 2 by Schroders), higher than the 1.6% available on US 10 year Treasuries.

5. Diversification benefit
What other bolt holes can weather volatile global equity markets, Brexit, US-China trade war?
Traders like volatility to a degree, but they try to diversify the risk so their portfolios aren’t fully exposed to the market's unpredictability. So buying notoriously safe bonds is a risk-averse strategy. You'll lose some money for that safety, but that loss can constitute ‘over-performance’ if other asset classes tumble. When risky assets sell off, negative yields on government bonds may be overshadowed by their proven ability to rally during recessions. Even at negative yields, bonds still are important. Figure 3 by Schroders  proves the 3-year rolling correlation between German Bunds and the German DAX stock market index. The two strong negative correlated, if you overlook the period between 2014 and 2017 when the European Central Bank’s (ECB) quantitative easing raised higher bond prices (lower bond yields) and higher stock prices.

Bonds  perform well when equities don't, vice-versa. In times of distress, you must reduce risk like this! If you are running for safety,  capital leave stocks to government bonds. Even if negative-yielding bonds lower your return compared to normal conditions, they can still reduce portfolio risk.
6. Roll down the yield curve later.
You're assuming the bondholder is holding it long term, but many don't.
In periods of uncertainty, demand for bonds increases, which makes their price increase. So you can profit from buying and selling government debt. If you think this  volatility will last, buy bonds now in the hopes that more investors flock to them later. You don't even need fellow investors to buy them! The central banks may be the ones to buy bonds to try to stimulate the economy.
Bond buyers can take advantage of the yield curve’s slope, which still can be steep even for negative-yielding bond markets in Germany and Japan. For example, a trader can buy a negative-yielding 3-year bond and sell it after a year. Since debt prices inversely correlate with yields, all else being equal, the value of the 3-year bond should be higher than, for example, a 2-year bond.
So long as shorter-term bonds' interest rates are more negative than their longer-dated counterparts,  the long-term bond's price should rise closer to maturity. But they can profit from this “rolling down the yield curve” as a short-term strategy, and must sell the bond well before maturity, because the bond will trade only at par when it expires.
For example, the Swiss 10-year bond was yielding -0.1% in Jan 2009. But at 22 August 2009, it had returned 5.6%! Negative yields can still positive return! Of course, this works both ways. Any rise in yields can make you lose money.
7. Some institutions are legally required to buy bonds! Liability matching with negative yields
The other reasons are about speculation or managing risk. But institutions can be legally mandated to buy bonds with negative yields.
Certain institutions must follow rules dictated by the banks, pensions funds or insurance companies who provide the bulk of the capital. Those rules mandate the managers to invest just in certain things like investment-grade bonds. Regulators force some clients to buy certain assets, like banks can only buy liquid assets.
Liability-relative investors, such as insurance companies and pension funds, don't care the absolute return or yields from bonds. They often buy bonds to “match”  their liabilities. The present value of these liabilities is calculated in many ways, but factors in government bond yields. For example, in Sep 2009 in the euro area, insurance industry used negative rates up to the 11 year maturity point (Source: EOPIA. See data from 31 July 2019). These liability-driven market participants can buy negative-yielding German Bunds to match a future liability . Their values will move in tandem with each other.  If they don't buy negative matching assets, they'll be exposed to significant risks if interest rates fell further.


Fiscal policy in the Dornbusch model I'm studying the Dornbusch overshooting model of the exchange rate. Specifically, I'm studying the model presented in a textbook by Copeland (2014).

The economy is represented by the following equations:

$y^d=h(e-p)$

$m_s-p=k\bar{y}-lr$

$\Delta p =\pi(y^d-\bar{y})$

$r=r^*+\Delta e^e$

$\Delta e^e=\theta(\bar{e}-e), \theta>0$

where $e$ is the log nominal exchange rate, $\bar{e}$ is the long-run value of the log nominal exchange rate, (∗) is the log domestic (foreign) price level, $m_s$ stands for log nominal money supply, $\bar{y}$ is potential log output, (∗) is the log domestic (foreign) nominal interest rate, $\Delta e^e$ is the expected change in the log nominal exchange rate, and ∆ is the rate of change for log prices.

You can simplify these equations into 2:

$p=m_s-ky+lr^*-l\theta(e-\bar{e})$

$\Delta p = \pi(h(e-p)-\bar{y})$

You then can find the long-run values of e, p, and q:

$\bar{q}=\frac{\bar{y}}{h}$

$\bar{p}=\bar{m}-k\bar{y}+lr^*$

$\bar{e}=(\frac{1}{h}-k)\bar{y}-k\bar{y}+lr^*$

What I'm confused about is the functioning of fiscal policy. If there is a permanent increase in government spending, $y^d=h(e-p)+g$, correct? From what I can gather, real shocks in the model won't lead to any changes in long-run prices. Indeed, if we plug this equation into the model, only $\bar{q}$ and $\bar{e}$ change.

I have seen an exam question which asks how fiscal policy might be used to bring $q, e, p$ to their equilibrium values after an increase in $r^*$. But surely this would necessitate a decrease in $\bar{p}$ (since $\bar{p}$ has increased with an increase in $r^*$), from which I gather isn't possible in the Dornbusch model?

There are some mistakes in your presentation of the Dornbusch model.

  $(∗)$ is the domestic (foreign) price level, $m_s$ stands for nominal money supply, $\bar{y}$ is potential output,

These actually stand for the log of price level, log of money supply and log of potential output. In international macro an unwritten rule is that lowercase letters represent logs of uppercase letters. For example, money supply is $M$ and $m = \ln M$. To be consistent you cannot add to the Dorrnbush model just government spending but you should add log of government spending. So $g$ instead of $G$:

$$y^d=h(e−p)+g$$

  From what I can gather, real shocks in the model won't lead to any changes in long-run prices. Indeed, if we plug this equation into the model, only $\bar{q}$ and $\bar{e}$ change.

The permanent government spending can in principle affect the price level through $\bar{y}$, but it depends on the background assumptions of the model. In Copeland, exchange rates and international finance (6th ed), you can see on pp207 that long run equilibrium values are derived based on arguing that in long run equilibrium $y^d =\bar{y}$. So in principle it could be affected by $g$ through $\bar{y}$ unless they are offset by changes in the other variables.

However, Copeland is not really clear on all background assumptions he makes in the model that he presents (at least not in the edition I have). For example, if we assume flexible exchange rate perfect capital mobility and the shock was completely unanticipated the $\bar{y}$ wont change in response to fiscal expansion as any change in $g$ will be just offset by changes to the exchange rate (due to capital inflows) and fiscal policy will only be able to expand output of its recipients at the expense of tradable sector in the economy. If some of the assumptions above are violated then the price level can actually change in the Dornbusch model.

In case you are interested in this for your own sake and you want to understand how price level responds to fiscal policy in general I recommend the following paper Devereux & Purvis (1990). Fiscal policy and the real exchange rate. They actually show that in the Dornbusch model fiscal policy can affect the price level and also discuss in greater detail how the sign of the effect depends on model parameters and assumptions of the model. If you are studying for an exam I would recommend clarifying this with your supervisor because Copeland does not provide detailed list of assumptions (at least not in the chapter 7.3 where he gives formal exposition of the model).


A confusion regarding search and matching model in Pissarides's "Equilibrium Unemployment Theory" book I was studying chapter 1 of the book "Equilibrium Unemployment Theory" and I got confused about the way Pissarides has defined the probability of a firm not finding a worker in a short time interval $δt$ as $1-q(θ)δt$  (page 7 of the book).

The first reason I got confused is that, to me that cannot represent the probability because if we take $δt$ large enough, the result can be negative.

Second, if we take $q(θ)δt$ as the rate (NOT the probability) of a firm finding a match in a time interval $δt$, then the rate of the firm not finding a match in the same time interval shouldn't be $[1-q(θ)]δt$ instead of $1-q(θ)δt$?

Any help would be much appreciated.

$q(\theta)$ is defined as the job-filling rate. Note that market tightness $\theta$ is not necessarily constant over time (Pissarides makes a dynamic analysis at some point). It may help to denote it $\theta_t$.
As an approximation, $q(\theta_t)\delta t$ is a probability for a firm to meet a worker between $t$ and  $t+\delta t$ for $\delta t$ small enough.

1) If you choose a large $\delta t$, this approximation is not valid anymore. In other words, assuming a large $\delta t$ prevent you from interpreting $q(\theta_t)\delta t$ as a probability. To define an equilibrium, Pissarides will anyway take the limit $\delta t\to 0$.

2) $q(\theta_t)$ is a rate (that is not constrained to be lower than 1), whereas $q(\theta_t)\delta t$ is a probability. Thus, the probability of a firm not meeting a worker between time $t$ and $t+\delta t$  is $1-q(\theta_t)\delta_t$.

$1-q(\theta_t)$ has no clear interpretation (it can even be negative). I guess (but I would like confirmation) that the rate of a firm not finding a worker cannot be defined in this case (or it would be $+\infty$).


What prevents a bank from simply going into their computer system and adding some zeros to their bank account? This question has been bothering me for a long time and I’m hoping that someone here can help to answer it.

I will use Greece as an example, but my question could really be applied to any country or even a bank:

While a government in the Eurozone, the Greeks for example, might have an actual building in which Euro coins are minted and Euro banknotes printed, these are most likely tightly regulated, with lots of physical checks to ensure that the Greek government doesn’t just print more whenever they want to.

I understand that physical money typically makes up a small portion of the actual money supply, so I assume that “printing them selves out of debt” regardless of whether it would be legal, isn’t an actual option for them.

However, what prevents the Greek national bank from simply going into their computer system and adding some zeros to their bank account? What system prevents any bank anywhere from doing the same? Does the bank of international settlements in Basel keep some kind of record of the total global money supply? How is this kind of fraud prevented?

I’m very curious to hear the answer!

Double entry book keeping.

If we take the example given here, of the Greek Central Bank (bank nerd trivia - interestingly the GCB is a commercial bank, listed on the Greek Stock Exchange), arbitrarily adding 000's to its deposit account.

This cannot be done as stated. The double entry book keeping accounting system on which all banking is based, requires that two ledgers are simultaneously updated, one with a credit and one with a debit. So if the GCB credits its own deposit account (which in practice would be a liability income account of some kind), there has to be a matching debit on another ledger. Those are the rules.

Typically deposit accounts increase their value either by people depositing cash in them (see below), by direct transfer from another deposit account, or by receiving a loan.

For example, if the government prints physical cash then this can be deposited at the central bank:

[debit cash account, credit GCB bank deposit account]

and this is how the US TARP intervention was performed. The federal reserve then used the money to buy loans from the US Banks. Note that a debit to an asset account increases its value, and a credit reduces it, whilst on the liability (right hand side), a debit reduces the account, and a credit increases it.

If the European central bank makes the Greek Central bank a loan then on the Greek side, the book keeping is:

[debit cash, credit interbank loan]

The GCB has received cash, and now has a debt that it owes the ECB among its liabilities.

The one thing it can't do is just arbitrarily increase the value of a single account, one of the main reasons for double entry book keeping is in fact to prevent that.

Bank fraud and the various forms of banking system abuse that do occur typically involve manipulations of banking operations within the framework of double entry book keeping. So while DEB. prevents accounts from having zero's tacked on the end, it doesn't prevent Mr. Smith the bank manager, lending his very good friend Mr. Brown a large sum of money that Mr. Brown has no intention of repaying.


What would happen if, hypothetically, money velocity would reach infinity? If, hypothetically, money velocity would reach an infinite speed, what would happen to that economy? If it would be infinite then it would mean that there would be infinite transactions and therefore Y would be infinite aswell? But infinite transactions would also be a sign of infinite demand and therefore the price would become infinite? Furthermore, if the money velocity is infinite, then it also would mean that every economic agent of the economy is in possession of the same amount of money and thus no wealth inequality would exist? I am curious to if there is a more solid theoretical answer to this and or what others think about this question.

No $Y$ would not become infinity. $Y$ represents the real output of an economy. A real output is limited by the economy's production possibility frontier. For example, if we assume simple production function $F(L)=10 \sqrt(L)$ with assuming the economy's stock of labor being $100$ such economy would not be able to produce more than $100$ units of output whether velocity is infinite or not.
In the short run an economy might not be using its resources to the full extent and in that case increase in the velocity could have some effect - although causality would go through prices not directly from velocity (see Blanchard et al. Macroeconomics an European Perspective).  Furthermore, to see the effect of velocity we could use a simplified model for money market equilibrium given by the equation of exchange:
$$MV=PY$$
Where $M$ is money supply, $V$ velocity, $P$ price level and $Y$ output. As explained above $Y$ is limited by the production capability of an economy. Hence $Y$ can be treated as fixed. Next $M$ is under a control of a government through central bank or other institutions. Since in your problem you dont specify any government action regarding $M$ we can also treat it fixed. Then the only other variable that remains is $P$. Hence if $V$ wold increase to infinity $P$ would also increase ceteris paribus.
However, as alluded to above in a short run economy might not be operating at its full potential. In such scenario increase in $P$ in the short run would also stimulate firms to produce more according to standard macro AS-AD model. Hence in short run some of the increase in velocity could also be soaked by $Y$ (through increase in $P$ induced by increase in $V$) but certainly not anywhere near to reach infinity. $Y$ must in the end respect the country's PPF which in a world of scarcity will be finite.

Furthermore, if the money velocity is infinite, then it also would mean that every economic agent of the economy is in possession of the same amount of money and thus no wealth inequality would exist?

No, a velocity of money is defined as: "the frequency at which one unit of currency is used to purchase domestically- produced goods and services within a given time period" (see FRED). High velocity does not imply more equal distribution of money. For example, imagine an economy with two people A and B, with A having one 1 $\\\$1$ banknote and B having 2 $\\\$1$ banknotes - and lets for simplicity assume anytime they spend money they instantly get it back as an income. If A uses his banknote only twice to purchase goods and services (he purchases something - immediately gets it back as an income and purchases something again) but B uses both of his two banknotes 10 times the velocity will be $22/3 \approx 7.33$. If B decides to use his banknotes 50 times the average velocity will be 34 and so on we could go on. Point is that higher or even infinite velocity does not necessary tell us by itself much about distribution of money in economy.
Furthermore, in public economics wealth is usually defined as a value of net assets. The value of net assets a person has does not correspond to the amount of money one has. For example, imagine person with house worth $\\\$ 100,000$ and no debt but absolutely no money. An economist would say that person's wealth is $\\\$ 100,000$. Now imagine person B with $\\\$1,000,000$ in a bank account while having also $\\\$1,000,000$ debt. Such person would be recorded as having $0$ wealth. Different scholars might disagree about whether wealth should include things like social security and on other details but I know of no economist that would equate wealth to money holdings.   Hence there is no simple relationship between distribution of money and wealth distribution.


If strict convexity of indifference curves isn't assumed, does MRS have to be negative? Strict convexity is defined as

  Let $X$ be a convex set in a real vector space and let $f: X\rightarrow \Bbb{R}$ be a function. $f$ is called strictly convex if $\forall x_1 \neq x_2 \in X,$ and $\forall t \in (0, 1)$:$$f(tx_1+(1-t)x_2) < t f(x_1)+(1-t)f(x_2)$$

If this doesn't hold, does marginal rate of substitution still have to be negative?

The marginal rate of substitution is the slope of the indifference curve multiplied by -1. Since the indifference curve is negatively sloped (as soon as the person prefers to consume the goods), the MRS is always positive. This has nothing to do with the convexity of the preferences.

If the indifference curve is convex without being strictly convex, then it is linear, i.e. it has constant slope. In that case, the marginal rate of substitution is constant (but still positive!).


Yield To Maturity exact meaning From investopedia: "YTM is the total return anticipated on a bond if the bond is held until it matures"

I have also seen this interpretation used in other sources however this seems wrong to me.

More specifically the I don't see how the anticipated rate of return can be independent from the cost of capital.

What I get from the math is that YTM expresses the cost of capital required for the net present value to be zero.

If the YTM is low then I can probably use a higher discount rate when calculating the present value of the bond. But using a higher discount rate than the YTM, by it's definition means a lower present value and therefore a negative net present value.

If the YTM is high enough then I will most likely use a lower discount rate, meaning a  higher present value and therefore a positive net present value.

Is my understanding correct?

The yield to maturity is the internal rate of return on the bond assuming no default. Given a series of promised cashflows in the future, to quote Prof. John Cochrane, "... the yield is just a convenient way to quote the price."

In the case of a zero-coupon bond with price $p_0$ and face value $c_t$ due $t$ periods in the future, the yield to maturity is the fictional, constant interest rate $y$ that solves the equation:

$$ p_0 = \frac{c_t}{(1 + y)^t} $$

More generally in the case of a non-negative series of payments $(c_1, \ldots, c_T)$, the yield to maturity is the fictional, constant interest rate $y$ that solves:

$$ p_0 = \sum_{t=1}^T \frac{c_t}{\left(1 + y\right)^t}$$

Given a series of promised coupon payments and the bond's face value, listing either a bond price or bond yield conveys the same information.

The yield to maturity does not tell you the expected return because you don't know the probability of default, recovery rate, etc....

Problems/errors with Investopedia's definition

The Investopedia definition you quoted is flawed because it's missing the critical phrase, "assuming no default." I've seen other sloppy, incomplete, or confused definitions on Investopedia and prefer Wikipedia.

Problems with your statements on the cost of capital

The yield to maturity has an unambiguous, widely agreed upon definition and is easily computable. On the other hand, the cost of capital is a more nebulous concept. Different practitioners or academics often use somewhat different definitions for the cost of capital and estimate these somewhat different notions of the cost of capital in different ways.

If you define the cost of capital as the return expected to be earned by investors, then yield to maturity is not the cost of capital over the period because the yield to maturity ignores the probability of default. (That said, it appears common practice in corporate finance to just ignore default probabilities and use yields as the cost of debt.)


Can a monopoly INCREASE the market surplus compared with a competitive market? Monopolies are often blamed for DWLs(Dead Weight Losses), while competitive markets believed to work without DWLs (assuming zero taxes/subsidies and zero externalities).

But I think I found an example when ABSENCE of a monopoly (in this case - a regulated natural monopoly) and presence of a competitive market can lead to smaller market surplus (i.e. DWLs). Am I correct? If not - why?

Let's assume there is a competitive market for a specific medicine. The competition is tough. Economies of scales aren't exploited to their maximum because in this case, only one firm would be able to survive (there is enough demand only for one Gigafactory) and for this reason, the state puts limit on maximum size of factories in order to prevent full exploitation of economies of scale.

So many people can't afford the medicine despite fierce competition. And if you don't buy anything, then obviously you can't have surplus. Not to mention, that in their totality the producers have lower surplus due to higher costs and lower sales.

Now let's suppose that one day the state decided that maybe a monopoly isn't so bad thing if it's properly regulated. So the state let the firms increase their exploitation of economies of scale and soon only one firm was left. Of course, it was regulated in order to prevent abuses. Now the market surplus increased. The surplus of consumers increased because now more people can afford to buy the medicine. The surplus of the producer's also greater than the total surplus of producers of the existed competitive market because the production of the medicine is less costly and it has more buyers.

But if we increased the market surplus compared with the existed competitive market, then what does it mean? Shockingly, it means that said competitive market had DWLs! Even if there were no taxes/subsidies and externalities.

UPDATE 01:

Ok, there is my try to formalize my thought.

Let's suppose that there is a perfectly competitive market for a particular medicine. There are NO taxes/subsidies or externalities.

The aggregative demand is Qd=1000-P

The aggregative supply is Qs=-2+0.2*P

P = 835 is the equalibrium price, thus we will have Q=165 as equalibrimum quantity.

The total surplus of consumers is equal to (1000 - 835)*165*0.5 = 13612.5

The total surplus of producers is equal to (835 - 10)*165*0.5 = 68062.5

Thus the market surplus is 13612.5 + 68062.5 = 81675

Now let's replace the competitive market with a regulated natural monopoly that can fully exploit economies of scale.

The aggregative demand is the same, it's Qd=1000-P

P=ATC=200 by the law, while the price without dead weight losses is P=183.(3)
It means that Q=800.

The monopolist's MC=-100+5*P

The total surplus of consumers is equal to (1000 - 200)*800*0.5= 320000

The total surplus of the monopoly is equal to 180*800*0.5+800*(200-180)=88000

Thus the market surplus is 320000 + 88000 = 408000.

As you can see, the total market surplus is bigger. Much bigger.

UPDATE 02

I found book "Intermediate Microeconomics" by  John Hey, it seems to support my conclusion. And there is nothing radical. I will quote some places from the chapter 29.

"One very obvious reason why a single large firm might be more appropriate in some industry is simply that a single large firm might have access to a more efficient technology than lots of small firms. This is a plausible case if centralisation of the industry has consequences in terms of
increased efficiency – contrariwise, if splitting it up into lots of little units causes losses of efficiency and increased bureaucratic costs. ... Even if it behaves like a monopolist – and there is an associated deadweight loss as a consequence – the end product for both consumers and producer might be better."

Under standard assumptions (some of which you state in your question: no externalities, etc.), no. This follows from the First Welfare Theorem.

Perhaps there are departures from standard models that would support something resembling your conclusion, but my guess is that most economists would view any such departure as the absence of “perfect competition”.

It might be a good exercise for you to write this down in a model (even a simple one) to help you figure out what those departures might be.

Regarding the example you put in your edit:

First, your "monopoly" is not a monopoly in the sense that economists understand it. Monopolies have market power, and thus have no supply curve. The firm in your example has a supply curve, and has no market power.

Second, the example you give does not seem to all support your argument. The firms in the two different situations you present seem to have markedly different production technologies (reflected in the differing supply curves). Your argument requires that the firms have the same technology, only that perfect competition prevents the exploitation of "economies of scale" of their (common) technologies. In other words, this result is driven by differences in production technology, and not from the fact that one market is perfectly competitive and the other is monopolistic. (I believe this is what some comments to your question mean when they say that this is a weird comparison.)

Moreover, where exactly is economies of scale in the second situation? How am I supposed to be able to interpret what is going on as the firm being able to take advantage of economies of scale? Bear in mind that in the second situation, you would have the same outcome (with respect to surplus) if there were many firms, as long as the aggregate supply curve were the same. The fact that there is a single firm is not at all important to the result you've found.

Let's illustrate this previous argument specifically in your example. It seems the correct intervention for the state is not to allow the monopoly to continue but to regulate it. Instead, what the state should do is to make the monopolist's production technology available to other firms that want to enter the market to foster competition.

Suppose that the state does this, and another firm enters the market with the same technology. Let's also suppose now that the market is perfectly competitive. Each firm's supply curve must be

$$ q_s = -100 + 5P $$

which makes industry supply

$$ Q_s = -200 + 10P. $$

Given the demand curve $ Q_d = 1000 - P $, the equilibrium price and quantity is

$$ P^* = \frac{1200}{11}; \, Q^* = \frac{9800}{11}. $$

Total surplus is then $$ \frac{1}{2} (1000-20)\cdot \frac{9800}{11}, $$ a $7 \%$ increase in surplus. Surplus should also increase further when more firms enter.


Why does a lack of pipelines cause Canadian crude oil to trade at a discount compared to US crude oil? There is an ongoing political battle in Canada about the proposed expansion of the Kinder Morgan Trans Mountain pipeline.

One argument in favour of the expansion that is repeated by just about every news article on the topic (example) is that the lack of pipeline capacity to get Canadian oil to market is causing Canadian oil to sell at a discount compared to US oil.

I'd like to understand this connection between pipeline capacity and oil price.

Naively, I would think that if pipeline capacity is scarce and the oil has to be transported by more expensive means like rail, its price would be higher to reflect the increased cost.

If you take the oil price as a given, meaning not set by Canada but by the market, then the costs of more expensive transport will eat the profit margins. For example, if the price of the oil is 100 USD per barrel, 20 USD would be costs for Canada, while for US producers the cost is, for instance, only 10$.

Keep in mind that a project investment analysis should take in account the cost of the pipeline divided by the number of years that it will be used to deliver that oil, adjusted for potential changes in oil demand, and including costs of insurance and removal at the end on the life of the pipeline. When all is taken in account, it might be that the oil will always be sold at discount when coming from Canada, and building the pipeline is not such a good investment after all - but only a full analysis would tell.


Benefits of a weak national currency I often hear that developing economies can benefit from a weak national currency. For example China as exporter deliberately makes its national currency weaker, so that it gains an advantage. Are there some underlying mechanisms which can explain this?

It's usually considered a short term solution when you would rather do something quick than engage in structural reform. Economics consider this to be a very poor way of increasing competitiveness, since you're effectively harming the purchasing power of your own citizens. Let me explain.

Say a business in the People's Republic of China has to pay its workers in its local currency, and can get most resources locally, so also paid in local currency. On the other hand its finished product is sold abroad, in a foreign currency.  What can the government do to help this business make money? It can go and buy vast amounts of the foreign currency on the local money markets (i.e. from all the businesses that do export). To get this money, it will pay the holders of foreign currency a slightly higher rate in the local currency. The more money it wants, the higher the rate. This of course, by basic supply and demand logic, devalues the local currency. This means it is easier for the company to pay its workers and local suppliers. The workers and suppliers won't be able to buy as many foreign goods, but if they buy local products, they won't be as much affected.

Now the government has a giant pile of money (you can think of it as savings extracted from the workers through the currency manipulation) and a lot of suddenly competitive exporters. To prevent other countries from easily doing the same, it will go and prop up the foreign currency (for instance by buying government debt at low rates).

Why would a country go through the trouble of doing that? Several reasons.
1. The country is dominated by an oligarchy of company owners and their friends and relatives, who want their businesses to do well, and don't care if this makes the ordinary workers relatively worse off. The ordinary workers must have no effective political representation for this to work, of course.
2. Your long-term policy slowly (or quickly) drives foreign businesses into bankruptcy, allowing you to corner an industry and effectively reap in large profits in that sector in the future, especially if initial set up costs are significant in terms of physical capital or intellectual know-how.

So why is this a poor solution long-term? Eventually, even the dumbest foreign governments will start to notice what you're doing and begin to impose punitive tariffs, or devalue their own currency in turn. Besides, doing it for decades will make your currency worthless. Italy's pre-euro currency, the Lira, got devalued so much, that Italian people wisened up to it, and asked for yearly salary increases to match. To keep its exports competitive, the devaluation rate had to be increased, and thus inflation rose and before you knew it, you had 500,000 lira banknotes. Taking the devaluation option for decades made it difficult for the devaluing country politicians to effectively implement the kind of structural reforms needed to make the country genuinely more competitive, discouraged savings, encouraged foreign currency hoarding by the population, depressing its long-term growth potential.

Whether a giant economy like China will be more successful with this policy by driving its competitors' companies bankrupt due to the sheer scale and sophistication of its operations is an open question. They have a lot of very smart people handling this, and as long as their markets such as the US do not retaliate and as long as inflation is in check, they're good.


Where does overseas money from exports go? I am a bit confused about foreign exchange reserves.

I was reading this website (https://www.thebalance.com/foreign-exchange-reserves-3306258#targetText=Foreign%20exchange%20reserves%20are%20the,to%20manage%20their%20currencies'%20values) which talks about the export money, banks and the central bank.

So when exporters from a country (let's say from Australia) recieve money (let's say USD), does that money go to a local bank? If so, does the local bank convert the money into AUD, or does the money go to the central bank which converts it into AUD? And does the overseas money just sit there in the reserve bank? Sorry if this is confusing.

There are two scenarios depending on the invoice currency of the trade.

If the trade is invoiced in USD, then the exporter receive USD from buyer, deposit this into the local bank. The local bank increase the amount in the exporter's bank account (in local currency), and put put this into central bank's reserve.

If the trade is invoiced in local currency, then the buyer has to exchange USD into local currency (similar process as above), then pay the exporter with local currency.


What's the term for consumers not being able to pay their mortgage? Is there a term for not being able to pay your mortgage(s)? Like "personal bankruptcy", "consumer default" or something like that?

Not paying a mortgage is first known as delinquency (when payments are missed) and later, default (when the loan is written down and the lender seeks foreclosure). But an inability to pay has no particular term of art attached to it, because an inability to pay can arise either out of illiquidity (debts exceed liquid assets) or insolvency (debts exceed liquid assets and ability to borrow against future income).


Production under uncertainty Currently I am trying to figure out some old problems for an upcoming micro exam. One of them is about production under uncertainty. The exercise seems standard to me, yet I am not sure whether I have the correct solution. The question is as follows.

A firm produces goods with the cost function $C(x) = cx^2$ where $x$ is the good. The price $p_x$ is exogenously given. The utility function of the producer is given as $u(y) = \sqrt{y}$ where $y$ is the income due to profit. Determine the optimal amount of produced goods $x$ in case i.) $p_x$ is deterministic and (ii.) the price is random where it is $p^0_x$ with probability 0.5 and $p^1_x$ with probability 0.5. How will the output change if there is some future price $p^f_x > \mathbb{E}[p^s_x]$?

My solution is as follows. First, we have that the firm's profit is

$$G = p_xx - cx^2$$

and its utility

$$ u(y) = \sqrt{p_xx - cx^2}$$

Now under deterministic prices the firm owner wants to maximize expected utility, hence

$$\frac{d\mathbb{E}[U]}{dx} = (p_xx - cx^2)^{-\frac{1}{2}}(p_x - 2xc) = 0$$

and therefore $$x = \frac{p_x}{2c}$$

In case of random prices the optimality condition is now

$$ \frac{d\mathbb{E}[u]}{dx} = \frac{1}{2}((p^0_xx - cx^2)^{-\frac{1}{2}}(p^0_x - 2xc) + (p^1_xx - cx^2)^{-\frac{1}{2}}(p^1_x - 2xc) = 0$$

right? After that I have to solve for $x$ and I got the result.

Finally, if there is some future price which is higher than the expected price I am not quite sure how to proceed. Intuitively, I'd say that if the owner would produce more goods but how would I show that in an analytic manner?

Under random prices, the producer solves

$$\max_x E[U(G(x,p)]$$

Compacting $U(G(x,p) \equiv U$, the f.o.c is

$$E[U'\cdot (p_x-2cx)]= 0 \implies x^* = \frac 1 {2c}\cdot \frac {E[U'\cdot p_x]}{E[U']} \tag{1}$$

or

$$x^* = \frac 1 {2c}\cdot \left[E(p_x) + \frac {\text{Cov}(U', p_x]}{E[U']}\right] \tag{2}$$

Now,
$$U' = \frac 1{2(p_xx - cx^2)^{1/2}}$$

and $U'$ is decreasing in $p_x$. Then (see for example Egozcue, Cogent Mathematics (2015), 2: 991082), Theorem 2(2) p. 6)

$$\text{Cov}(U', p_x] <0$$

So we conclude that

$$\implies x^* < \frac {E(p_x)} {2c}$$

the RHS being the risk-neutral solution.

Plugging the distributional assumption for $p_x$ in $(1)$ gives us an implicit function for $x^*$. If one is patient enough to go through some boring algebra after that, one will obtain a very simple nice looking closed-form solution.

As for the last question, it is poorly worded and it does not make much sense, or it is a trick question, since the set up implies that the producer must commit to an output level before the price is actually realized. If this was not the case, then the producer would have no reason to solve an expected utility maximization problem, but he would wait to see the actual price and optimize accordingly.

If the last question alludes to a dynamic maximization problem, then this is a whole new situation, and additional assumptions and modelling are needed in order to proceed.


Perfect Bayesian Equilibria What are the perfect Bayesian equilibria in the following game?

The part that is confusing me is the right-hand side, where player 2 is of type N.

Thanks!

EDIT: My workings out so far are given below:

We start off by noticing that if player 2 is type $N$, it can be rational for him to play both $Steal$ and $Split$, as he does not know what player 1 plays.

Let us therefore first look at the case where player 2 plays $Steal$ if he is type $N$. Then player 1's expected payoff from stealing is $100\beta$ and his expected payoff from splitting is $50\alpha + 50\beta$. In this case player 1 steals when $100\beta > 50\alpha + 50\beta \Leftrightarrow \alpha < \beta$, and therefore splits if $\alpha > \beta$ and is indifferent when $\alpha = \beta$. This means we get the following potential PBE's in this case:
\begin{align*}
    (Steal, Steal; \alpha \leq \beta) \text{ and } (Split, Steal; \alpha \geq \beta).
\end{align*}

Let us next look at the case where player 2 plays $Split$ if he is type $N$. Then player 1's expected payoff from stealing is $100\beta + 100(1-\alpha-\beta)=100(1-\alpha)$ and his expected payoff from splitting is $50$. In this case player 1 steals when $100(1-\alpha) > 50 \Leftrightarrow \alpha < 1/2$, and therefore splits if $\alpha > 1/2$ and is indifferent if $\alpha = 1/2$. This means that we get the following potential PBE's in this case:
\begin{align*}
    (Steal, Split; \alpha \leq 1/2) \text{ and } (Split, Split; \alpha \geq 1/2).
\end{align*}

As such, we have can write up the set of pure-strategy PBE:
\begin{alignat*}{1}
    \text{PBE} = \{&(Steal, Steal; \alpha \leq \beta),(Split, Steal; \alpha \geq \beta), \\ &(Steal, Split; \alpha \leq 1/2),(Split, Split; \alpha \geq 1/2)\}.
\end{alignat*}

Hint:
You can convert the game into the following normal form:
\begin{equation}
\begin{array}{|c|c|c|}
\hline
& Steal & Split \\\hline
Steal & 100\beta, 0& 100(1-\alpha),0 \\\hline
Split & 50(\alpha+\beta),50+50(1-\alpha-\beta) & 50,50 \\\hline
\end{array}
\end{equation}
Note that Steal is a weakly dominant strategy for player 2, so (Split, Split) can never be an equilibrium.


The government does not need to gather capital via tax and Treasuries to spend? According to MMT I have watched Warren Moslers interview. https://www.youtube.com/watch?v=W97s3zbFKvc&t=841s. At 7:26 he says that the Treasury does not issue securities for the purpose of financing spending (it is actually accomplished by the Fed debiting the Treasury and crediting somebody else). And at about 10:20 - 11:00 he says that it is not the governments problem if the Treasury notes are not bought. This logic is very unlike the usual understanding that the Treasury issues bonds to finance government spending.
He says they are spending first and only after that the bonds get sold.
But what if nobody buys the bonds then it is a problem, right? Am I understanding this correctly? Is this common knowledge? Because it sounds very different from what you read about elsewhere.
My questions: Does the government really not need taxes and debt to spend since it can just print money? Are taxes really only for reducing money supply. Are Treasuries only for controlling interest rates?

There are a number of points in this question, and I do not want to write too long a response that goes too far into opinions. I will note that I am a proponent of Modern Monetary Theory (MMT).

I will first address two questions. I will write from the MMT perspective, and note that I am referring to a sovereign that controls the currency it borrows in (examples: the United States, Canada, Japan). Countries that borrow in a foreign currency (or a currency it does not control, like the euro bloc), face the same financial constraints as a household or firm.

  Does the government really not need taxes and debt to spend since it
  can just print money?

Technically, the government does not "need" to tax to spend, but that ignores a side effect that is also part of MMT: taxes create a demand for state money, and without taxes, government spending would be highly inflationary. (Some countries might be able to achieve the feat of abolishing taxes by having large natural resource endowments exploited by a government-owned enterprise, but I will ignore that.)

A less controversial way of stating this is that a central government does not need to tax before spending, which is undoubtedly true.

As for bond issuance, MMTers observe that spending is implemented by the government issuing cheques (or making bank transfers), which has the effect of creating "reserves" (bank settlement balances at the central bank) in the banking system. It can do this (within institutional limits) without prior borrowing. Bonds can then be issued, which eliminate these reserves that are created.

  Are taxes really only for reducing money supply. Are Treasuries only
  for controlling interest rates?

Although taxes eliminate "reserves", that is not their primary role, rather it is inflation control.

Treasuries drain "reserves," and allow for interest rates to be able the interest rate paid on excess reserves (which used to be zero in the United States).

This post by Bill Mitchell (one of the co-founders of MMT) discusses the lack of necessity of government debt issuance. Link to website. This subject is also covered in Chapter 21 of the textbook Macroeconomics by Mitchell, Wray, and Watts.

I will then turn to:

  Is this common knowledge?

I do not think there is a way of avoiding opinions on this question. In my view, the number of informed dissents to the MMT view in 2020 are very few, the issue is almost entirely how the statements are worded. Conventional economists would refer to this discussion as "money-financed expenditure," and the practical effects of this under conventional theory are hard to distinguish from MMT's description. There is a debate about the inflationary effects of such a policy.

  He says they are spending first and only after that the bonds get
  sold. But what if nobody buys the bonds then it is a problem, right?

The statement "nobody buys the bonds" is probably better expressed as: can a floating currency sovereign default on its debt? That is a difficult question, as the institutional regime in each country is different. The MMT argument is that any institutional arrangements that could lead to a default are self-imposed, and could be eliminated by the government whenever it wishes. I think you would need to ask a separate question on that topic for a country of interest.


How did the silver standard "insulate" China from the beginning of the Great Depression (1929-1931) There is a segment from this article that I want to unpack:

The Great Depression was a global crisis—almost. Every significant
economy was devastated, with one notable exception: China. The reason
was simple. In 1929, the US and every other major nation pegged their
currencies to gold. As the economic historian Barry Eichengreen has
described, adherence to this standard punished countries by imposing
“golden fetters" that led to crippling deflation. The fixed exchange
rates of the gold standard helped transmit the monetary shocks around
the world.

Clearly, the silver standard didn't help China in the later years, but I'm keen to understand how China's economy fared so well in these specific years: 1929-1931. When I read the above, I'm not exactly sure what the logic is. My best hunches were:

Price of gold increased faster than the price of silver because gold is more of a "haven" asset, and thus providing China with more competitive exports
Or, the peg to silver itself wasn't the main factor, but merely being pegged to something different than the other countries which helped China avoid the transmission effects

But after looking at historical prices of both metals leading up to and following the Great Depression, I saw divergence, but not dramatic divergence. So my intuition has been depleted.
Question
How did the Silver Standard insulate China from the broader economic contractions?

As argued by Ho (2014) the silver standard helped China because downward trending silver prices were producing mild inflation during the period 1929-31 and also helped to devalue exchange rate between gold and silver.
First, as already discussed in this Economics.SE answer, mild inflation is important for an economy to avoid excessive unemployment due to nominal rigidity of wages especially in the recession. In fact this is one of the many reasons why the conventional wisdom of combating recession is expansionary monetary policy (see Blanchard et al. Macroeconomics a European Perspective).
Furthermore, it is worth noting that Great Depression was so severe in US because US monetary policy went in exactly  opposite direction to the above mentioned conventional wisdom. As famously argued by Friedman and Schwartz in the monetary history of  the United States the Great Depression would arguably be only mild recession if Fed would not decide to pursue disastrous contractionary monetary policy.
Second, as discussed by you and also in the article you linked increasing gold prices and decreasing silver prices allowed the exchange rate depreciate vis-à-vis gold standard currencies   and currency depreciation is another conventional way of countries fighting recessions (again see Blanchard et al mentioned above). Normally, this policy can be risky because other countries could also competitively devalue their currency leading to potential currency wars, but in this case as their currency depreciated gold standard currencies appreciated.
Also, I disagree with your assessment here:

But after looking at historical prices of both metals leading up to and following the Great Depression, I saw divergence, but not dramatic divergence. So my intuition has been depleted.

According to data from macrotrends the price of gold rose between Jan1929-Dec1931 from approximately ${\\\$}315.38$ to ${\\\$}402.71$ (growth rate of $\approx +27\%$), while  price of silver decreased from ${\\\$}8.07$ to ${\\\$}5.17$ (change of $\approx -35\%$) in the same period. Or in another terms the ratio of prices $Gold/Silver$ changed from approximately $39$ to $78$ (approximately doubled) - that is quite a large change in exchange rate. I would be surprised if such appreciation of Gold vis-à-vis Silver would not result in some non-trivial boost for Chinese exports.


What country has the highest requirements for loans? I was looking to find countries with the toughest regulations on loans(general loans not any specific ones)

By that I mean:
 I mainly care to know where it is toughest for people to get a loan from a bank.

To consider is:

•   where requirements are highest for people to qualify for a loan

•   where banks do the most thorough checks in a persons financial status (like SCHUFA in Germany)

•   where do banks have to pass the most audits and tests to prove their business is legit

I was looking at IWF Data but could not really get that sort of info from there.

I am basically looking to invest in Banks and want to find the best country to do that based on their banking infrastructure. I want to find lenders that make SURE their clients pay by setting the bar very high (I know that doesn’t mean they won’t default anyways)

A good starting point would be the "getting credit" subindex of the Doing Business rankings compiled by the World Bank.  There is also an explanation of the methodology.  This is supposed to be a test of the business environment, i.e. rules laid down by the government and central bank rather than about the behaviour of market participants.


Why is South Korea classified as an Emerging Market economy? I have recently found out that South Korea is part of the MSCI Emerging Markets Index, and that the South Korean Won (KRW) currency trades as a non-deliverable currency. Furthermore, the South Korean Interest Rate Swaps are also non-deliverable and trade with a relatively wide spread, signalling poor liquidity.
This came as a total surprise, because when I used to think of South Korea, I used to think of companies such as Samsung, Kia, Hyundai or LG (i.e. globally competitive, tech-savvy corporations). Also, the South Korean capital Seoul came to mind, which somehow has an image of a modern, western-type capital city.
For many years, I somehow had a mental image of South Korea as a tech-savvy country, along the lines of Japan, and certainly a developed economy.
As I am no expert on Asia and do lack deeper economic education, I wanted to ask for an economic explanation as to why South Korea is still an emerging economy, and also, what are the country's prospects of becoming a developed economy?
Thank you so much for any type of explanation, including perhaps a historical one which would explain why South Korea is economically where it is today.

Economists already consider South Korea to be developed economy. In fact some scholars already argued that Korea was a developed economy already sometime in late 90s - 2000s (see Kim, 1988; Kim, 2011). Opinions might differ because there is no precise definition of what developed country is but quite often in literature this is proxied with whether economy reached high income status and South Korea already did that by World Bank definition some time ago.
MSCI index has its own classification that does not necessary reflect what economists think about the country. MSCI is a private company and they create their own classification in order to help people who invest to decide how to invest. The MSCI classification framework can be seen here. The classification is based on economic performance but really the numbers and thresholds are all arbitrary and do not even refer to economic literature or literature on economic development. For example, other indexes such as  Financial Times Stock Exchange
(FTSE) index already reclassified South Korea as a developed country as you can read here.
You should not take classification by different indexes  too seriously, or give them too much weight, they do not necessarily reflect economic literature. If you would start your own private company doing the same business as MSCI you could decide to classify even Maldives as developed country or USA as a developing one. This being said the indexes do this to help to guide their investors so they will try not to do it completely without any thought (and go to such extreme as I suggested in the previous sentence) but in the end it is their private decision, and different indexes will decide differently.


What would be the economic impact of disallowing the renting of housing? I'm just trying to think through the implications of the theoretical scenario of disallowing the renting of housing.
My goal would be to reduce wealth inequality - as rent is generally a wealth transfer from the poor to the rich - but this would pull on so many threads in the economy that I wanted some other opinions.
So a builder or investor could own an apartment complex and sell the individual apartments, and the buyers could pay in installments that would be essentially the same as renting, from the perspective of the person living there, with the difference being that after 10 or 20 or 30 years the occupant would own the apartment and not owe anything but taxes and maintenance.
If someone knows that they are only going to stay someplace for a year or two then the buying/selling process would need to be much simpler and cheaper to avoid all of the transaction costs and paperwork.
How to distinguish apartments from hotels is another question.

If the required downpayment is substantial there will be fewer households if renting is illegal. This means the average number of people per household would be greater. Some people would be living in larger households as they save for a downpayment. If the required downpayment is zero then price volatility would imply that some home sellers have a debt from an earlier home ownership experience. If experience shows that debtors tend not to pay then the required downpayment will be substantial. The most likely outcome is substantial required downpayments and a greater average number of persons per household.


Why are Big Macs supposed to cost the same around the world? The Big Mac index is frequently mentioned in media, but I could never understand how it makes any sense. According to Wiki:

  The Big Mac Index is published by The Economist as an informal way of measuring the purchasing power parity (PPP) between two currencies and provides a test of the extent to which market exchange rates result in goods costing the same in different countries.

Essentially the index is trying to say that if a Big Mac is more expensive in Switzerland compared to India, then the Indian rupee is undervalued or conversely that the Swiss franc is overvalued. But why would a Big Mac cost the same in all countries? Labor costs vary widely between different countries and so do costs such as real estate. There are even significant differences within the same country - a Big Mac in central London will cost you more than a Big Mac in a highway restaurant.

So why is the index taken at face value by anyone? How does it measure anything except the cost of producing a burger in different countries?

Because of the Law of One Price, identical products are supposed to cost the same in different places. Since McDonald's applies standards for all the individual franchisees, the product is supposed to be identical even if it isn't perfectly identical. The law assumes there are no transportation costs. If lower costs exist in one place and there are no transportation costs you would move enough product until costs equalize. When the costs equalize the product price can equalize. This is not realistic for perishable products. The problems with this "law" are described in the limitations secton of the Big Mac Index entry in wikipedia.


Inflation reasons beyond Friedman I think everyone know this Friedman's quote nowadays:

  Inflation is always and everywhere a monetary phenomenon in the sense
  that it is and can be produced only by a more rapid increase in the
  quantity of money than in output

However, things are not that simple. The US had emitted lots of money and has very little inflation. In some cases there's also a spiral effect. Some heterodox economists talk about lack of supply of goods too. And emission still applies only to the mid & long term.

To sum up, the questions would be:

Can businesses be an important cause of inflation, or is it a pure monetary effect?
In not-so-developed countries, there might not be much competition.  Can that generate inflation?

Edited. Original question: Can anyone help me understand when that quote is true and when other factors should be considered too?

The US had emitted lots of money and has very little inflation.

Are we talking about quantitative easing here?  It depends how broadly you are willing to define inflation.  Quantitative easing has caused asset price increases, just not consumer price inflation.  Look at stock prices, which are about 50% overvalued according to some well regarded investors (e.g. Warren Buffett).

A common complaint about quantitative easing is that it is overly focused on investments and not focused enough on people (Wall Street over Main Street).  It has been successful at propping up stock prices to the point that they have returned to their 2008 levels, when they were regarded as being in a bubble.  Quantitative easing is heavily focused on investment assets.  It doesn't seem to be getting money into the hands of consumers.  Household incomes are still low.

In short, I don't believe that the current US recession has disproved Friedman's statement.  The Federal Reserve has eschewed its traditional open market operations in favor of quantitative easing, which has affected investments without increasing the money supply available to consumers.  A better question might be why quantitative easing is widely regarded as expanding the money supply when it doesn't seem to actually do that--at least not the kind of money supply whose expansion causes consumer price inflation.

Note:  I'm sure that there is theoretical work disagreeing with Friedman's statement.  I haven't tried to cover that.  I was caught up in the questionable assertion that the US money supply has expanded without causing inflation.  I suspect that this is too new for a consensus to have arisen. This is my thoughts on one possible explanation.


A trillion dollar invention Is it possible to make a trillion dollar invention or exploration? I mean an invention or exploration that will be valued at trillion dollars.

Sure, because dollars are nominal goods. Perhaps a ticket to Star Wars Episode CXVIII will cost one trillion dollars. If you meant today, without huge changes in the price level:
 Yes, there is way more than one trillion dollars of wealth in the world. Depending on the laws of physics and psychology there is probably something you would be able to market for this amount. (Private luxury homes in other dimensions.) Coming up with it is another question, but there is no law of economics that makes this impossible.


Difference between Engel curve and income expansion path I was scrolling through many questions on this site about income offer path and it appeared to me that income offer path, income expansion path, engel curve all are different terms with same meaning, i.e. graph of the engel function.

  Can somebody explain what is the difference between these terms, if any?

Both of these curves describe the same phenomenon, the change in consumer choice as income changes. The difference is in the displayed variables. The Engel curve is a relationship between the consumption of a good $x$ and income $I$, whereas the income expansion path (IEP) shows the relationship between the consumption of a good $x$ and a good $y$ as income increases. These are the coordinate systems the curves are usually displayed in, so Engel in $(x,I)$, IEP in $(x,y)$.

The Engel curve would show you if good $x$ is inferior.

One can also learn this from IEP (it would have a negative slope somewhere), and in case both goods are normal IEP would show you if the consumption of one good would increase at a faster rate than that of the other.


Why absolute value in elasticities and marginal rate of substitution? This is a point I find very confusing and very hard to justify to students. Depending on the books, one finds many different conventions regarding the sign of elasticities and marginal rate of substitution (MRS). Some define them with absolute value, some don't, and  one sometimes finds inconsistencies inside a single book or set of notes.

My questions are :

To your knowledge, what is the most conventional position regarding the use of absolute value in the definition of

Own-price elasticity
Cross-price elasticity
MRS

Is it mere convention or is there somewhat of a rationale for taking absolute value in some/all/none of the cases?

I think there are pedagogical advantages to discussing both the raw numbers and the absolute values and I think the benefits of both explain why they both show up (sometimes in the same text, even).

Each elasticity number gives two bits of information. First, the absolute value with respect to 1 and second, the sign. Now, clearly, if you had a negative elasticity, you could compare it to -1. However, it becomes somewhat difficult to teach when using phrases like "greater than" or "less than" -1 to discuss a good being (in)elastic, since "greater than -1" is actually inelastic if the elasticity is negative. It is much more intuitive to be able to discuss the ratios of percent changes if "greater than" does in fact mean that the top is bigger than the bottom and vice versa for "less than".

Of course, there is also a bunch of information tied up in the sign of the elasticity. We get the Law of Demand out of own-price elasticity, we get compliments/substitutes from cross-price elasticity, etc. So it is important to still make sure students understand the importance of the sign.

When I am teaching, I try to discuss both parts explicitly, but make clear that the elasticity itself includes the appropriate sign. I think most books are trying to capture these two bits of information in one way or another. In any case, the formal definition of elasticity should include the sign, but if one is just talking about how elastic a good is, the absolute value could be reported (with the note that it is the absolute value of the elasticity, not the elasticity itself).

As for MRS, it's usually not the absolute value, per se, that we report, but rather the negative of the derivative dy/dx. This is quite standard, since it has the intuitive interpretation of the consumer being willing to give up so many units of x for so many units of y. Since indifference curves are usually convex, this derivative is negative, thus changing the interpretation (and intuition) somewhat if we don't negate it.


Equilibria in Signaling and Screening I am now studying the Spence's job market model using both signaling and screening.

It turns out that, when a worker's reservation value = 0 (she can get 0 if she does not get employed), then both pooling and separating equilibria always exist in signaling. However, in screening, pooling equilibrium does not exist, and the range for separating is also narrower than in signaling.

Why is that so? Does it have anything to do with the refinements such as intuition criterion?

Is it true that screening will always be stricter than signaling?

(Sorry for being verbose - still quite new to this area and do not know how to describe the problem easily.)

(Reference: The question is discussed in the 13th chapter of MWG's textbook.)

The difference between signaling and screening stems from the fundamental difference in bargaining power- who offers the contract for which her utility is the highest. While in screening the uniformed party proposes the contracts, in signaling it is the informed party.

For your general conjecture, I refer to Stiglitz and Weiss (1990): "Sorting Out the Differences Between Signaling and Screening Models":

  However, if the indifference carves of the informed agents satisfy what roughly would amount to a single crossing property in two dimensions, and some technical conditions hold, then all contacts in the screening game break even, and the set of outcomes of the screening game is a subset of the outcomes of the corresponding signaling game.

It has nothing to do with refinements such as the intuitive criterion, divinity and so on. These refinements are useful to "refine away unreasonable equilibria". Here, the issue is that the set of PBE is potentially quite rich while some equilibria rely on "unreasonable beliefs". The refinements formalize which of those beliefs are "unreasonable".

If you are interested, you can also look at settings that combine signaling and screening. This is the literature on informed principals, initiated by Myerson and Maskin & Tirole (1990, 1992). The informed principal first offers a contract to screen the agent while the contract proposal itself may reveal private information of the principal.


Uncovered set versus top cycle set (voting theory) I am struggling to see when, given a policy set  $X = {x_{1}, … , x_{m}} $ , with $m > 1$, its uncovered set and top cycle set may not coincide (i.e. when the top cycle set has policies that are not in the uncovered set).

The uncovered set is defined as follows. Policy $x$ covers $y$ if (1) it is majority-preferred to $y$, and (2) all policies that are majority-preferred to $x$ are likewise preferred to $y$. If there is no $x$ that satisfies criteria (1) and (2), then we say that $y$ is uncovered.

Next to this, the top cycle set is defined by partitioning the policy set into k disjoint subsets, $${L_{1}, … , L_{k}}, \text{where} \; 1 \leq k \leq m $$ by iteratively applying the idea of the top cycle set, where
$$(L_{1} = {x ∈ X | \; ∄ \; y ∈ X / L_{1} \; s.t. \; y \succ x})$$

We call these subsets of $X$ levels to suggest a mental picture: one can see the electoral environment as a series of levels or plateaus (if $k > 1$). Each plateau electorally dominates those below it: each policy at a given level is majority-preferred to all policies at lower levels. Further, every policy at a level covers all policies at lower elevations. This implies that $X$’s uncovered set is a subset of $L_{1}$, the uncovered set of $X / L_{1}$ is a subset of $L_{2}$, and so forth.

Defined constructively in this setting, a policy is in the top cycle set ($L_{1}$)  if and only if it is reachable from every other policy by a chain of straight majority preference.

Specifically, given the above definition I cannot understand why the above definition of the top cycle set implies that $X$’s uncovered set is a subset of $L_{1}$.

If anyone could think of an example or some other explanation/input for this I would greatly appreciate it. Thanks in advance!

This example comes from Miller (1980) (the paper that introduced the uncovered set definition).

Suppose that the majority's preference is defined by:

\begin{aligned}
x \succ y \\
x \succ z \\
y \succ v \\
y \succ z \\
v \succ x \\
z \succ v
\end{aligned}

In this example, we have the top cycle set $L_1 = \{x, y, z, v \}$ (you can check this).

But notice that $y$ is preferred to $z$ and all policies that are preferred to $y$ (namely $x$), are preferred to $z$. So, $y$ covers $z$ and the uncovered set $U$ is such that $U = \{x, y, v\}$ and $U \subset L_1$.

Among other stuff that Miller proves in his paper, we have the following three:

if there is a Condorcet winner, $L_1 = U$;
if $L_1$ has three elements, then $L_1 = U$;
$U \subseteq L_1$.


Why does a firm in a perfectly competitive market not need to cover fixed costs in the short run? While studying, I came across the solution that in some cases only the produces surplus, PS, needs to be positive, however not the profit, $\pi$. That means that if $PS \geq 0; p \geq AVC$, whereas, e.g. in long-run, $\pi \geq 0; p \geq AC$ to supply a positive amount.

I wonder, why the fixed costs need not to be covered in some cases (e.g. $R(Y) \geq VC(y) \Leftrightarrow PS \geq 0)$.

Your question is somewhat wrong in the sense that a firm needn't cover its unavoidable costs (sunk costs). If there are fixed costs that are somewhat avoidable or recoverable (for instance selling an old machine for a salvage value), we need to cover them.

Now coming back to the main question. We can express the profit of a firm using the following equation
$$\pi = R-VC-FC$$

Assume that a firm has already incurred an unavoidable FC of \$5000. The VC that the firm is incurring is \$2000 and the revenue (R) of the firm is \$2500. If the firm continues to operate, its net loss from the business will be \$4500, however, if the firm ceases its operation, its loss would amount to \$5000. Thus as long as a firm has a positive contribution ($R-VC$), the firm must continue its operations.

All in all, Microeconomics says that there is no use crying over a spilt milk. If you have already incurred a sunk cost, it is better to get whatever you can.


Consumer theory with lump sum fee Please look at only part ii.

I wrote budget constraint $x_1+x_2=10-f$

When I don’t pay fee, my optimal value values${}=(1,1).$

In the case of free disposal, $b_1=b_2=5$ but I consume only $x_1=0$ and $x_2=5.$

However I couldn’t find the value of $f.$ How can I find?

Consumer's problem in default scenario:

\begin{eqnarray*} \max_{x_1, x_2} & \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} & \ x_1 + x_2 \leq 10 \\& \ 0 \leq x_2 \leq x_1  \end{eqnarray*}

We can easily show that the solution will satisfy $x_2 = x_1$, and therefore we can rewrite the above problem as:

\begin{eqnarray*} \max_{x_1} & \ \frac{x_1}{(1+x_1)^2} \\ \text{s.t.} & \ 0 \leq x_1 \leq 5   \end{eqnarray*}

Solving this problem we get $x_1= 1$. Consequently, $x_2 = 1$ and the utility of the consumer in optimum is $\dfrac{1}{4}$.

  Consumer's problem with a possibility of freely disposing good 1 after paying the fee:

\begin{eqnarray*} \max_{x_1, x_2} & \ x_2 \\ \text{s.t.} & \ x_1 + x_2 \leq m \\& \ 0 \leq x_2 \leq x_1  \end{eqnarray*}
where $m$ is the net income of the consumer after paying the lump-sum fee.

Solution to this problem is $\left(\dfrac{m}{2}, \dfrac{m}{2}\right)$, and the utility of the consumer in optimum is $\dfrac{m}{2}$. Comparing this with the default scenario yields that consumer is indifferent between the scheme and the default when $m = \frac{1}{2}$. Therefore, the maximum willingness to pay to dispose the good at $0$ price is $10 - \frac{1}{2} = 9.5$.


How to prove that a point is a maximum point I have the following function:

$$
\Pi =\int_{0}^{z}[x_{1} + \alpha y + \alpha \frac{N-2}{2}y - \beta(z) - \gamma ( \beta(z) - \beta(y))](N-1)y^{N-2} dy $$

The first derivative with respect to $z$ is:

$$
\frac{\partial \Pi}{\partial z}=[ x_{1} + \alpha z + \alpha \frac{N-2}{2}z - \beta(z)] (N-1) z^{N-2} - (1 + \gamma) \beta'(z)z^{N-1}
$$

at $z = x_{1}$ it must be zero so that:

$$
[ x_{1} + \alpha x_{1} + \alpha \frac{N-2}{2} x_{1} - \beta(x_{1})] (N-1)  - (1 + \gamma) \beta'(x_{1})x_{1} = 0
$$

The solution is:

$$
\beta(z) = \frac{(N \alpha + 2)(N-1)}{2(N + \gamma)}
$$

What I want to prove is that the function $\Pi$ have a global maximum in $x_{1}$. Until now I think I've only obtained a necessary condition. Can you suggest me a way to prove it?

EDIT: $\beta(\cdot)$ is an increasing function.

The first order condition results in a differential equation in $\beta(x_1)$,

$$\left[ x_{1} + \alpha x_{1} + \alpha \frac{N-2}{2} x_{1} - \beta(x_{1})\right] (N-1)  - (1 + \gamma) \beta'(x_{1})x_{1} = 0$$

$$\implies (1 + \gamma) x_{1}\beta'(x_{1}) = -(N-1)\beta(x_{1}) + (N-1)\left(1+a \frac{N}{2}\right)x_1$$

or generically $$g(x_1) y' = f_1(x_1)y + f_0(x_1)$$

Where $y \equiv \beta(x_1)$ and the other obvious mappings.

This has general solution

$$y= \beta(x_1) = C\cdot  \exp \{F\} + \exp\{F\}\int \exp\{-F\} \frac{f_0(x_1)}{g(x_1)} dx_1, \;\;\; F\equiv \int \frac{f_1(x_1)}{g(x_1)}dx_1 $$

and doing the work results in

$$ \beta(x_1) = \beta(z^*) = Cx_1^{-(N-1)/(1+\gamma)} + \frac{(N \alpha + 2)(N-1)}{2(N + \gamma)}x_1$$

I guess there exists a condition in the model that leads to $C=0$ so that we get the linear expression for the $\beta$ function

$$ \beta(z^*) = \frac{(N \alpha + 2)(N-1)}{2(N + \gamma)}x_1$$

Regarding whether $\Pi(z)$ has a global maximum at the stationary point $z^* = x_1$, working the 2nd derivative with respect to $z$, and evaluating it at $z^*=x_1$ is what one can do.

Then, if I haven't done any trivial mistake, it appears that a necessary condiiton for the second derivative to be negative at $z^* = x_1$ is that $\alpha(1+\gamma) <1$ and, given this, the condition for a global maximum at $z^*=x_1$is

$$N  > \frac{2+\gamma}{1-\alpha(1+\gamma)}$$

which appears to put a lower bound in the number of participants in the auction, which I guess is what $N$ measures.

But this is the 2nd time I had to guess (and a third guess is that parameters are positive), validating @denesp comment that even though the OP issue is mathematical, providing context and information on what the various symbols stand for, is needed if one is to have a good-quality question to work with. After all, this context is what would make this an economics question and not a math question.


How rate rise by the Fed, higher than market expectation will lead to strengthening of Dollar? If the Fed follows through on its forecasts, though, and raises rates faster than markets expect in 2016, the dollar may well rise further, dampening inflation quickly

From : http://www.economist.com/news/finance-and-economics/21679806-first-three-pieces-federal-reserves-imminent-interest-rate-decision

What is the logic behind this? Thanks :)

A higher than expected interest rate in the US would result in a higher than expected interest rate differential with other currencies. This will drag more money to the US as investors will try to benefit from this higher interest rate. Moreover, a faster tightening may create the perception that the Fed is more confident about the economic environment. This good news for the US can translate in dollar appreciation.

An appreciated currency makes imports cheaper, which negatively affects inflation.