Why does Denmark have such a huge disparity between its income and wealth Gini coefficients? (After-transfer) income inequality and wealth inequality are obviously not the same thing, but I would intuitively expect that, all else equal, they would be reasonably well correlated across countries. Looking at the lists of countries ranked by income inequality and by wealth inequality, this appears to be more or less the case - with one glaring exception. As of the year 2000, Denmark's income Gini coefficient of 0.226 was the lowest listed in the OECD, and one of the lowest on the world (third-lowest, although that's as of 2011) - but its wealth Gini coefficient of 0.808 was the third-highest in the world.

Is there something unique about Denmark's economy that explains the gigantic disparity between its income and wealth inequality?

The disparity exists because a high income taxes level income inequality but do little to fix wealth inequality.

  Wealth inequality is always higher than labor income inequality, due to income originated from wealth (or capital income), such as rents, dividends or royalties and the increase of asset values [6]. This source of income accounts for a large portion of the national income (˜ 20%-35% [11]), hence its substantial effect on wealth inequality. Additionally, though labor income accounts for most of the national income produced in all western economies, only a relatively small fraction of it contributes to accumulating personal wealth due to taxation and spending. [...]

  There is an obvious positive correlation between wealth and income inequality. However, it is a characteristic of mainly poor countries. The correlation between the Gini indices for the middle-high and high levels of income is 0.23, while for the low and middle income countries it is 0.68.  [...]

  Mathematically, increasing the average tax rate reduces the disposable income, hence effectively reduces savings. However, the relatively limited range of average income tax rates makes its effect on wealth inequality relatively low, in practice. Therefore, it can be concluded that progressive income tax is likely to be an impractical measure for affecting wealth distribution, as suggested previously [20, 25, 36, 52]. This non-trivial effect of income on wealth inequality is originated from the imperfect correlation existing between income and wealth [18, 21, 23, 24]. When income tax is increased, the top earners, who are not necessarily the wealthiest individuals in the population, have a larger difficulty of accumulating wealth, with respect to the wealthiest. On the other hand, it barely affects the wealthiest individuals. Therefore, such an increase might even deepen the wealth gap. When decreasing income tax, the opposite effect will therefore occur. In addition, progressive taxation, which might have a significant effect on the distribution of income, will have a small effect on wealth inequality. Therefore, we conclude that affecting income inequality or changing the average income tax in order to effect wealth inequality are likely to be ineffective means. An evidence for this can be found in the economies of Denmark and Sweden, two of the most equal countries in terms of income in the world, with a high tax burden, in which wealth inequality is among the highest within the OECD countries [53].

If I can find out why wealth inequality is so high in Denmark, I'll add more details about that, but the above address why one should not really expect super strong correlation between wealth and income inequality, which I take it to be the main question here.

What I could find out quickly is that wealth inequality in Denmark continued to rise, as of 2015.

  The new figures reveal that the poorest 10 percent in Denmark had average assets of 211,000 kroner in 2015, 13,000 kroner less than the year before, while the wealthiest 10 percent owned average assets of 6.76 million kroner – an increase of 375,000 kroner compared to 2014. [...]

  The figures revealed that Denmark’s richest 10 percent now own as much as 70 percent of the rest of the population put together – a clear indication the government didn’t need to consider any more reforms aimed at benefiting the wealthiest Danes.

This actually gets even more intriguing because until 1998 Denmark had one of the highest wealth taxes.

There might be an explanation with the data from the graph above, which is from the early 2000's, because pension funds were not included in wealth back then, as calculated by the Danish tax authority:

  The Danish wealth data are considered of a very high quality, and they have been fruitfully
  used to study retirement savings (Chetty et al. 2014), intergenerational wealth mobility (Boserup
  et al. 2014), and the accuracy of survey responses (Kreiner et al. 2015). The data does have two
  limitations, however. First, they exclude funded pension wealth before 2012, because such assets
  were not subject to wealth taxation. [...] Second, there is a break in the data in 1997 when the wealth tax was abolished. After 1997, while the Danish administration continued to collect wealth data from third parties, it stopped
  asking households to self-report assets not reported by third parties.

I guess this emphasizes why wealth inequality is a lot harder to measure (than income inequality). If "very high quality" wealth data has those kinds of issues...

Anyway, here's a more detailed analysis of wealth inequality trends in Denmark:

  Figure 1 shows wealth shares in three broad classes: the bottom 50%, the next 40%, and the top
  10%. Measured in this way, wealth inequality has been relatively stable in Denmark over the last
  three decades. Throughout the period, the bottom 50% of the distribution owns a tiny fraction of
  aggregate wealth: their assets are barely higher than their debts. Therefore, almost all wealth is
  owned by the richest half of the population, and it is shared about equally between the middle
  40% and the top 10%. While the wealth shares in the figure are overall stable, wealth inequality
  did increase somewhat from the mid 1980s to the early 1990s. During this time, the top 10% wealth
  share grew while the bottom 50% wealth share shrank. This evolution was driven by the dynamics
  of asset prices, in particular housing prices, which fell significantly during this period. Because the
  share of housing in asset portfolios tends to be decreasing in the level of wealth, housing slumps
  hurt the bottom more than the top, leading to a rise in wealth inequality.

So if you want to know the source of the wealth inequality in Denmark, you have too look further back in history, and unfortunately there's apparently not much good data for that.

The same paper goes on comparing the rise in wealth inequality with the US, and finds that in Denmark it was raising more slowly, especially in the top 1% and top 0.1%. They also find that dropping the wealth taxes didn't change this pattern, however, they attribute this to an "equalizing force":

  One important confounding reason why wealth inequality has stabilized in Denmark (despite wealth tax cuts) is
  likely to be the sharp rise of pension wealth, from around 50% of national income in the late 1980s to 178% in 2014.
  Because pension wealth is relatively equally distributed, rising pension wealth tends to reduce inequality. As shown by
  Chetty et al. (2014), automatic employer contributions to retirement accounts increase pension and total wealth substantially
  for middle-class Danish households.


Why is a fire service considered to be a non-rival service, while a highway is considered to be a rival good? We consider highways to be an example of an impure private good because while it is non-excludable, it is a rival good. That is, another person's consumption (usage) of the highway affects my usage of it.

In that same sense, wouldn't a fire station also be rival, at least in theory?

For example, if a fire stations has five trucks for a locality, and 6 fires break out at separate places, why can't we consider this service as a rival service?

One common definition of a non-rival good (see here and (A)) is a good for which the consumption of additional units involves zero marginal social costs of production. On this definition the service provided by a fire service would always be rival, because (even with plenty of capacity in terms of fire engines and staff) each visit to attend a fire involves some extra cost (eg vehicle fuel).

More loosely, a good may be said to be rival if consumption by one person prevents simultaneous consumption by others (this can also be found here), implying that a good is non-rival where that is not the case.  In this looser sense, infrastructure such as highways and services such as the fire service might be held to be non-rival in circumstances where capacity is large enough that consumption by some rarely prevents or interferes with consumption by others.

For those who adopt this looser definition, the difference between a highway network and a fire service, which may lead to the former but not the latter being characterised as a rival good, relates I think to the likely level of supply or capacity.  Given high levels of vehicle use with peaks at certain times (rush hours), and given the costs of road-building and other demands on land use, it isn't likely to be practicable to provide enough highways in the right places to make traffic jams a rarity.  On the other hand, given that a fire at any one location is a rare event, and given normally a high degree of independence between fires at different locations, it should be possible to provide sufficient capacity in a fire service that the scenario of fewer fire engines than fires will rarely occur.

Reference

(A) Nicholson W (9th edn 2005)  Microeconomic Theory: Basic principles & Extensions p 596.


Arrow’s impossibility theorem and voting schemes Arrow’s impossibility theorem states that there is no procedure for aggregating individual preference orderings into a collective preference ordering that satisfies certain apparently desirable axioms. This has often been taken to mean that there is no such thing as a good voting system. Actual voting systems, however, do not typically attempt to generate a (complete) social ordering. The goal, rather, is to simply select one candidate who will govern — and the task of the ranking the others, while perhaps interesting, is not of immediate practical importance. This raises a simple question: does Arrow’s result extend to preference aggregation procedures that choose a single candidate based on a profile of individual preferences?

To make this more precise, let me (informally) restate Arrow’s axioms as adapted to the problem at hand:

Unrestricted domain. The procedure chooses a best candidate for every possible list of individual orderings.

Weak Pareto. If all voters prefer candidate A to candidate B, then candidate B is not elected.

Non-dictatorship. There is no voter who alone determines which candidate is elected (regardless of the other voters’ preference orderings).

Independence of Irrelevant Alternatives. If candidate A is elected when opposed to a set of rival candidates, then A will still be elected when opposed by some subset of this set of rival candidates.

(Notice that the transitivity axiom cannot, it would seem, be adapted to this context.)

Question: does there exist a voting scheme that satisfies all of the axioms listed above?

The answer is no in the following sense.

Suppose there are $N = \{1,\dots,n\}$ individuals, and $M$ alternatives, where $|M|\geq 3$.

Let $\mathcal{P}$ be the space of strict preferences over $M$, each individual has preferences $P_i \in \mathcal{P}$, and let $P = \{P_1,\dots,P_n\}$ be the profile of preferences.

We look for a social choice function $f:\mathcal{P}^N \rightarrow M$.

$f$ satsifies Unanimity (Weak Pareto) if all individuals rank $a$ first, then $f(P) = a.$
$f$ is Strategy-Proof if it is a dominant strategy to tell the truth about your preference.
$f$ is Dictatorial if there exists an individual $i$ such that whenever $f(P) = a$, then $a$ is ranked first according to $i$.

  Theorem (Gibbard–Satterthwaite) A social choice function $f$ is Unanimous and Strategy-Proof if and only if it is Dictatorial.


Debt-to-GDP or Debt-to-Revenue? Which is a better measure of the health of an economy between Debt-to-GDP and Debt-to-Revenue? Why?

You may want to consider this as the difference between the potential to pay and the actual means to pay on hand. Since government revenues will ultimately come from GDP (excepting foreign aid etc.), you will be certain that debt to revenue will be higher (smaller denominator).

How useful is this as a measure? I suppose, as with all things, it depends. Debt to GDP is a well understood and widely used statistic, and so it lends itself to comparison between other nations. Ultimately GDP is a pool from which the government, in principle, could raise revenues from.

In contrast, revenues are your means to pay right now. Debts generally don't all need to be paid at once, and so your means of paying right now may not matter as much as the fact you are able to pay them when they're due (after all, the money you have right now may be better put to use in some productive activity that will enhance your means of paying the debt in the future). If the bill collectors are at the door, suddenly your revenues matter quite a bit!

I suppose the closest I can get to a direct answer to your question might be that if you doubted the ability of a government to tap into these potential revenues (ie. people won't pay taxes), then debt to revenue would likely be more appropriate given that it is an accurate portrayal of the means to pay.

I'll close with a numerical example. Suppose there were two governments that were identical across all respects, only tax to cover their expenses and have \$10 billion in debt with two different payment plans. Country A must pay back \$1 billion a year over 10 years and Country B must pay back \$2 billion a year over 5 years.

Country A's debt to revenue would be 10, while Country B's debt to revenue would be 5. The numbers are different, but we haven't really learned much about the health of these economies.


What is the best observable proxy for the instantaneous risk-free market interest rate? I am exploring a number of different models using daily data, of the US economy, and am trying to gather up observable proxies for some of the most important macro and micro variables, and surveying methods of constructing nowcasting indicators for others. One of my goals is to produce a number of forecasts of what will happen today, i.e. in the next 24 hours, based on variables that are observable after we wake up in the morning, say between 6 and 9 AM. So I am looking for the best available market indicator of the interest rate that is observable at a moment in time, i.e. that is based on continuously reported transactions.

The best I have found so far is the overnight effective federal funds rate (EFFR). That is a big important number available at daily frequencies, but it is still a good ways away from the quality of, e.g., treasury bond yields, which are available pretty much minute-by-minute based on trades in a real market. And I worry that the EFFR may be fully or partially set by Fed policies, rather than private market forces. Can I do better than the EFFR, in terms either of frequency or market-based-ness? Should I be constructing the interest rate from the T-bill yields? What do people think is the best observable proxy or proxies?

Oh, and I should specify that I am looking for the nominal rate. I think I have some good daily proxies for inflation, so I'd like to keep these separate.

SOFR is becoming more and more popular
Also there are other measures of repo rates, and also so futures on sofr


How does a govt surplus increases investment? From the closed economy equilibrium condition $I = S + (T - G)$, how does exactly a surplus (which I assume would be used for paying debt) increases investment in the same amount? I ask that because if the govt pays $(T - G)$ - assuming a surplus - to the public with a marginal propensity to consume equal to c, savings (and thus investment) would increase in $(1-c)*(T-G)$ and not $(T - G)$ (which is, of course, bigger). Am I losing something here?

The increase in savings you described does happen, in the short-run. Remember, tho, investment and savings are flows. Accounting identities provide us truths about their ex-post behavior, not planned or desired savings.


What is the difference between trade deficit and current account deficit? I read this article and still don't understand the difference. "Spends more money on what it imports compared to the money it receives for what it exports" (how it defines current account deficit). "There is more being bought by the country than there is being sold" (how it defines trade deficit). Isn't this the same thing described by different words?

They are very related concepts but they are not completely same.

Current account deficit occurs when country spends more on imports than it receives in imports.

Trade deficit means that more imports are being sold than exports by a country.

A trade deficit is also a part of current account deficit as current account can be decomposed as:

$$CA=X-M+NI+NT$$

Where $X-M$ is the trade balance and and if $X-M<0$ we say country has a trade deficit, but as you can see the current account also includes net income which is income of country’s resident minus income paid to foreigners and NT is net transfers which includes things like foreign aid and remittances. A current account deficit occurs when $X-M+NI+NT<0$.

So although they are related and trade deficit is usually a big component of current account deficit they are definitely not same. It’s even possible for country to have current account surplus while having trade deficit.


The effects of Quantative Easing (QE) on FX rates? What are the effects of quantitative easing on the foreign exchange rates?

If the central bank starts buying debt denominated in domestic currency, I would think that the market would be flooded with domestic currency, making it less valuable with respect to other currencies.

Are there any other effects which I'm not taking into account?

Any help would be appreciated

You've identified the first-order effect: the supply of local currency has increased, so its price (and from the perspective of other currencies, that's its exchange rate) decreases. That's assuming that the move wasn't already (fully) priced-in by the market.

There are lots of other effects.

There will be a change in domestic interest rates. The particular rates affected, depend on which debt has been bought. And there will be knock-on effects in those markets. This may affect inward investment in those markets.

QE in itself sends a couple of signals to the market. Firstly, that the Bank sees deflationary pressures a year or two ahead (assuming its mandate is an inflation-rate target); and secondly, that it intends to apply inflationary pressure to deal with it. The market will react to those signals, depending on how far they diverge from what the market had already anticipated. Those reactions might include increased optimism or pessimism about future domestic demand, and / or about the health of domestic industries, and that can have an affect on exchange rates.

In short, the central bank is pressing levers that touch on much of the macro-economic picture, so there are lots of second-order effects. The nature and direction of these effects will depend on expectations before and after the announcement, as well as what's actually going on in the economy.


Population growth and household income I heard a real estate agent say “as a city’s population increases so will the household income.” He went on and gave an example of how a city had a population growth of 20% and a household income increase of 30%. He explained stated again that “whenever there is an increase in population there is an increase in household income.”

Can you please explain the connection between the two ?

Well this all boils down to the economic reason why cities exists at all.

The economic reasons for cities to exist are:

Division of labor - as was famously observed by Adam Smith “It is the great multiplication of the productions of all the different arts, in consequence of the division of labour, which occasions, in a well-governed society, that universal opulence which extends itself to the lowest ranks of the people.”

Virtually all economists agree that division of labor leads to higher productivity and hence incomes and the higher population of certain area is the more possibilities for specialization there are. In a city with 5000 people there will be less scope for specialization than in city with 5000000 people. This is so because for example in a small community there might not be enough demand to sustain some very specialized goods or services. For example, I play a lot of board games when I was living in a small city there were just two board games shops with just general selection of games. Now I live in big city and there are separate shops for table top games, separate shop for just warhammer figurines, another for mainstream board games, separate shops for indie board games etc.

Increasing returns to scale -  most firms will have them at least at a portion of their cost curve. More people - means that firms there can scale up their production and increase their profit. This in turn leads to higher demand for workers and also to higher wages. There will be also some macroeconomic effects no matter for which factor of production the returns increase.

transportation cost - Last factor connected to the population is transportation costs. If everything is produced and exchanged in one hub people and businesses incur lower transportation costs than if everything is produced in separate locations.

So to sum it up as population density of certain are increases incomes should increase as well. There is also a downside that as population density increases price levels increase as well due to higher overall demand, but as long as people are free to move they would only move to the city if the increases in income outpace price level increases so empirically you will observe that as population of urban area increases the real incomes will rise as well.


Find Pareto-optima two good economy There's an exercise I cannot wrap my head on. I actually have the solution to the problem, but still I don't quite understand it.

Let an economy be composed of two consumers $A$ and $B$ and two goods $X$ and $Y$, such that $Y$ is produced using $X$ with production function $Y = \sqrt{5X}$. We suppose $A$ and $B$ have identical utility function $U(X,Y) = XY$. Initial endowment of the economy is $X = 1$ and $Y=0$ Find the Pareto-optima of this economy, and express them as a function of $X_A$, the quantity of good $X$ allocated to consumer $A$.

So what I did to find those optima is to put $MRS_{XY}^A = MRS_{XY}^B$ subject to constrains $X_A + X_B \leq 1$ and $Y_A + Y_B = \sqrt{5(1 - X_A - X_B)}$. But it seems to me that I got to few equations to actually solve the problem. What I find is, putting $X = 1- X_A - X_B$:
\begin{align*}
Y_A = \frac{\sqrt{5X}}{1 - X} \cdot X_A
\end{align*}
but I can't seem to be able to go any further... In the solution, it reads:
\begin{align*}
\frac{Y_A}{X_A} = \frac{Y_B}{X_B} (=\frac{Y_A + Y_B}{X_A + X_B}) = \frac{\sqrt{5}}{2\sqrt{X}}
\end{align*}
without explaining how to derive the last equality.

The first equality, as you've already found, equates $A$'s and $B$'s MRSs. The last equality comes from setting the marginal rate of transformation $MRT_{XY}$ between $X$ and $Y$ to the common MRS. For a production function with a single input and a single output, $MRT_{XY}$ is equal to $MP_X$, the marginal product of the input good $X$. That is:

$$ MRT_{XY} = \frac{1}{2}\sqrt\frac{5}{X}. $$

For details, see Mas-Colell, Whinston and Green (1995), section 16.F.


Are there any datasets for protectionism? Are there any data sources or variables that measure how protectionist a country is? If so, what are they?

Looked a bit further, I think I possibly found what you're looking for:

The Overall Trade Restrictiveness Index (OTRI) summarizes the trade policy stance of a country by calculating the uniform tariff that will keep its overall imports at the current level when the country in fact has different tariffs for different goods.

The data:
http://siteresources.worldbank.org/INTRES/Resources/469232-1107449512766/OTRI2009.xlsx

Background: http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/0,,contentMDK:22574446~pagePK:64214825~piPK:64214943~theSitePK:469382,00.html

hope this helps,
best regards


How do you achieve quantative tightening? In order to achieve quantative tightening the US Government is letting debt expire and not renewing it.

What does this actually mean in practice?

Presumably once money is made it is essentially impossible to remove it completely from supply?

When a U.S. Treasury bond matures the principle is repaid in money. If the owner of the bond is the Federal Reserve, it can reinvest the money in an asset like a U.S. Treasury bond. If the Federal Reserve wants to reduce the money supply the receipt of the money is coincident with a reduction in the account of the federal government. Since it is computerized they do not have to destroy any currency and coins; they change a number in a computer.


Why is market equilibrium pareto efficient? Let us assume that the current price $P$ is lower than the 'equilibrium price' $P^\star$ so that $Q$ is lower than $Q^\star$. If we move from this combination towards the equilibrium one, it may be the case that the producer surplus decreases. I understand that the total surplus (consumers+producers) increases, but it is not clear to me why producers should be willing to reduce their surplus. So it looks like to me that there is not pareto improvement in moving from $(P,Q)$ to ($P^\star$,$Q^\star$), in that consumers are better off but producers may be worse off.
What is wrong with my reasoning? Could you help me with this, please?

A situation can be Pareto efficient without also being a Pareto improvement over every other situation.

So, here for example, equilibrium is Pareto efficient, but -- as you have noted -- is not a Pareto improvement over every other situation.

Example. Say I have 100 apples to divide between A and B. The allocation (60, 40) is Pareto efficient, but it is not a Pareto improvement over the allocation (59, 41).


Examples of Applied Micro Paper with R (!) Code and Data in Public Repository Can anyone suggest examples of relatively simple applied micro papers (e.g. an RCT or straightforward regression model) of possible interest to a general audience, that have their data and code, written in R, as opposed to Stata or anything else, in a public Internet repository?

That's a lot of conditions, but I'm hoping for examples that meet all of them. AER, AEJ: Applied, REStud and REStat all have nice public repositories. With the exception of REStat, which uses Dataverse, the repositories aren't searchable, so it's a pain to look through each set of files to see whether they use R or Stata. Most everyone, myself included, uses Stata, but I'd like to find some papers using R.

You could try searching the Harvard Dataverse for fileType:"R Data" like this: https://dataverse.harvard.edu/dataverse/harvard?q=fileType%3A%22R+Data%22

I think your use case of wanting to search for data in specific formats such as RData is a common one so I just created an issue about improving Dataverse to support this use case better: https://github.com/IQSS/dataverse/issues/2707


JK Galbraith Quote: How to interpret? I came across a very thought-provoking quote by JK Galbraith concerning supply-side economics. The quote is as follows:

  The poor don't work because they have too much money. The rich don't
  work because they don't have enough money. If you believe that, you'll
  believe anything. -- J.K. Galbraith

With famous titles like: The Affluent Society, Galbraith was a expert in economics. Also he is very witty, and between these qualities I'm having a hard time connecting the dots for myself. I'm not sure how much of his explanation/comment is sarcasm and how much is genuine.

Question: Is there any notion/assumption/principle in supply-side economics that could legitimately be alluded to in Galbraith's remark here?

In other words, he must have phrased it this way for a reason; and let's not forget, this is in the context of supply-side economics. So by juxtaposing these false statements with our intuitive understanding of supply and demand, it's as though he's mocking that school of thought. So what is the original concept of the supply-side school of thought that he his mocking?

Surely there is something he thought was paradoxical in supply-side theory that made him say this comment.

Here is the quote, rephrased:

This is a false statement: (A) the poor don't work because they have too much money.

This is also a false statement: (B) the rich don't work because they don't have enough money.

If you believe those false statements (A) and (B), you are very credulous.

The poor have a much higher marginal value of money than the rich, so their incentive to work, to earn more money, is very high. For the rich, it's the other way around.

Ken is making explicit two assumption that are fairly common (in that school of thought and some other Conservative groups), but rarely spoken so plainly, and rarely spoken together. When they made explict and are put together, it makes it much clearer that to hold both of them at the same time is absurd.


why is labor $h_t$ often average working time percentage of non-sleeping time in RBC? Cobb-Douglas production is often given by $Y=AK^{\alpha}L^{1-\alpha}$, and $L$ is often just labor. But in RBC model, I notice that labor $h_t$ is restricted to $[0,1]$, and interpreted as working-time perecentage of non-sleeping time. Is this necessary restriction for RBC? Or can we allow $h_t$ to be in $[0,100]$ (with utility function $U(c_t,100-h_t)$)?

Specifications of this form treat labor $h_t$ and leisure $1-h_t$ as two alternative ways of spending a total time endowment.  Labor can be used as an input to the production function, while leisure enters directly into the utility function as a desirable end in its own right.

In this environment, there are many equivalent normalizations. There is no need to say that the total time endowment is 1; instead, we could just as easily say it is 100, and by modifying the production function and utility function accordingly we'd get an isomorphic model.  There's nothing special about any particular normalization.

That said, given a particular functional form for utility $u$, it often does matter how we define the time endowment, and this issue has played an important and somewhat embarrassing role in the history of RBC. In particular, if we normalize the total "time endowment" to 1 and say that $h_t$ is the labor fraction of the time endowment, then it can matter greatly whether we define the time endowment to be 24 hours every day for all individuals, limit to non-sleeping time, limit to non-sleeping time of working age adults, etc.

Why does this matter? The basic RBC model has typically assumed that intratemporal utility depends on a constant-returns-to-scale composite of consumption $c_t$ and leisure $l_t$. Given this, we can infer that this composite must have a constant elasticity of substitution of 1 from the fact that long-run labor supply seems roughly stable even as productivity and real wages have a strong upward trend. (See, for instance, pages 6 and 7 of Prescott 1986.) An elasticity of 1 means that the composite is Cobb-Douglas in consumption and leisure, taking the form $c_t^{1-\phi}l_t^\phi$ for some $\phi$.

RBC models then generally assume a constant intertemporal elasticity of substitution $1/\gamma$ for this composite, giving us the period utility function found on page 6 of Prescott (1986):
$$u(c_t,l_t) = \frac{[c_t^{1-\phi}l_t^\phi]^{1-\gamma}-1}{1-\gamma}$$
In the common special case with unitary intertemporal elasticity of substitution $1/\gamma=1$, the above reduces to simply
$$u(c_t,l_t) = (1-\phi)\log c_t + \phi \log l_t\tag{1}$$
Now, one important feature of a dynamic macroeconomic model is the Frisch elasticity of labor supply, which is the elasticity of labor supply to real wages holding marginal utility of consumption constant. This is important because it determines how the household will intertemporally substitute labor in response to the intertemporal pattern of real wages and real interest rates.

At an optimum the marginal utility of leisure must equal the real wage times the marginal utility of consumption, so for constant marginal utility of consumption $\lambda_t$ we have the condition
$$\frac{\phi}{l_t} = w_t\lambda_t$$
It follows that the Frisch elasticity of leisure $l_t$ with respect to $w_t$ is $-1$; this is a more-or-less direct consequence of the assumption that consumption and leisure form a Cobb-Douglas aggregate. But since $l_t = 1-h_t$, the local Frisch elasticity of labor $h_t$ is
$$\text{Frisch elasticity of labor}=\frac{dh_t/h_t}{dw_t/w_t}=\frac{-dl_t/h_t}{dw_t/w_t}\\
=-\frac{l_t}{h_t}\times \underbrace{\frac{dl_t/l_t}{dw_t/w_t}}_{\text{Frisch elasticity of leisure}} = \frac{1-h_t}{h_t}$$
From this, we can see that the definition of "time endowment" 1 matters greatly. For instance, from Table 1 of the most recent American Time Use Survey we see that average working hours per day are 3.14. If we take the time endowment to be all 24 hours, then $h=3.14/24 \approx .13$, and the implied local Frisch elasticity is about 6.6. If, on the other hand, we take out sleeping and other personal care activities, the remaining time endowment is 14.5 hours, in which case $h\approx.22$ and the implied Frisch elasticity is about 3.5.

Both these values are far higher than most Frisch elasticities estimated using micro data, as discussed by Chetty et al. (2012). Moreover, it seems bizarre that something as deep and important as the Frisch elasticity should depend on a choice as arbitrary as the definition of "time endowment". This connection is ultimately attributable to the assumption that $c_t$ and $l_t$ form a constant-returns-to-scale composite; King, Plosser, and Rebelo (1988) show that if we discard this assumption, a more general "KPR" functional form is consistent with the stylized fact of stable long-term labor supply, and we can potentially calibrate this more general form to the (much lower) elasticities estimated in the labor supply literature rather than imposing an unrealistically high Frisch elasticity by fiat.

For reasons that I do not completely understand, much of the RBC literature has followed Prescott (1986) and used the "log-log" utility parameterization (1) above, rather than calibrating the KPR functional form to match some externally estimated elasticity. This is probably due to a combination of inertia and convenience: the high Frisch elasticity guaranteed by the log-log parameterization helps to make labor volatile enough to match the macro data. And this is all connected to a lengthy and convoluted debate about the possible difference between "micro" and "macro" elasticities - see Chetty's slides or paper to read more about this.


Solving a maximization problem by substitution when the constraint is in implicit form I am trying to understand how the first order conditions for an interior solution of a maximization problem were derived using the substitution method.

The problem is:
$$\max\limits_{x\ge0,y\ge0}P(a-x)+(1-P)(b-y)$$
subject to
$$Pf(x)+(1-P)f(y)=c$$
where: $a,b,c>0$, $P\in (0,1)$, $f:[0,+\infty]\to[0,+\infty]$, increasing and strictly concave over its domain.

I can see how this is solved using a lagrangian to find from the first order conditions that $f'(x^*)=f'(y^*)$. Strict concavity of $f$ then implies $x^*=y^*$. But I'm at a loss as to how we can solve it by substituting the constraint in the objective function. Since $f$ is invertible, if $y$ did not appear in the constraint I would find $x$ from the constraint by inverting $f$ and substitute it in the objective function. Doing this here leads to complications that seem unnecessary for this simple problem. There has to be a simpler way that I cannot figure out: what is it? Thanks!

Here are two methods. First method: the substitution can be made by inverting $f$. Since $f$ is strictly increasing and continuous, $f^{-1}$ is well-defined. The constraint can therefore be written
\begin{equation*}
x = f^{-1}\Big(\dfrac{c-(1-P)f(y)}{P}\Big)
\end{equation*}

The objective becomes
\begin{equation*}
\max_{y \geq 0}{P \Big[a-f^{-1}\Big(\dfrac{c-(1-P)f(y)}{P}\Big)\Big]+(1-P) (b-y)}
\end{equation*}

The derivative of this expression with respect to $y$ equals
\begin{align*}
& (1-P) f'(y) (f^{-1})^{'}(\dfrac{c-(1-P)f(y)}{P})-(1-P) \\
= & (1-P) \dfrac{f'(y)}{f^{'} \circ f^{-1}(\dfrac{c-(1-P)f(y)}{P})}-(1-P) \\
& = (1-P) (\dfrac{f'(y)}{f'(x)}-1)
\end{align*}
And thus $f'(y^{*})=f'(x^{*})$ at the optimum.

Second method: since $f$ is invertible, we can do a change in variables and define $w=f(x)$ and $z=f(y)$. The problem then becomes
\begin{equation*}
\max_{w \geq 0, z\geq 0}{P(a-f^{-1}(w))+(1-P)(b-f^{-1}(z))}
\end{equation*}
subject to
\begin{equation*}
P w +(1-P)z=c
\end{equation*}
Substituing $w=(c-(1-P)z)/P$ in the problem delivers
\begin{equation*}
\max_{z\geq 0}{P(a-f^{-1}(\dfrac{c-(1-P)z}{P})+(1-P)(b-f^{-1}(z))}
\end{equation*}
Differentiating with respect to $z$ yields the same solution.


Nested CES Production Function If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a, b together are complements of c, d together. How would such a production function look like?

I saw some people include a Cobb Douglas there to simplify the function. So, under what situations do people use Cobb Douglas in a nested CES? Cause the factor price of a here in my model will decrease asymptotically to zero, if use the classic CES, b will be completely substituted, which I don’t want.

Consider the CES production function over four goods defined as follows:
$$ x = (\alpha j^{\gamma} + k^{\gamma})^{1/\gamma}$$
$$ y = (\delta l^{\beta} + m^{\beta})^{1/\beta }$$
$$ z = (\zeta x^{\xi} + y^{\xi})^{1/\xi }$$
If $\gamma$ is near one then $j$ and $k$ are near perfect substitutes. If $\beta$ is near one then $l$ and $m$ are near perfect substitutes. If $\xi$ is a large negative value then the composite goods $x$ and $y$ are near perfect complements


What is the difference between Herfindahl Index and the Concentration Ratio I was recently reading about the Herfindahl Index and from what I've learned so far, is that HHI is preferred over the Concentration Ratio. However, I didn't quite understand the reasoning that lead to this conclusion. What impact does squaring have on the final result? Any hints or suggestions for further reading will be appreciated.

From the Industrial Organization by Belleflame and Petiz (Page 34/35, Chapter 2):

  While the m-firm concentration ratio adds market shares of a small number of firms in the market, the so-called Herfindahl index (also known as Herfindahl–Hirschman index) considers the full distribution of market shares.

We can conclude that the mathematical approach in HHI is more adequate for a full market setup, rather than observing a particular concentration ratio in one firm.
Also from the same book

  the Herfindahl index provides a better measure of concentration as it captures both the number of firms and the dispersion of the market shares.

Hence the squared market shares.

For your information, there is another concentration measure called the Lerner Index, although the Lerner index is a snapshot of the intensity of competition. You can calculate L by finding the difference between price and marginal costs as a percentage of the price. More formally:
$L=\frac{p-C'}{p}$

But it is easily observed that L ignores some dynamics of the markets. Lower prices do not necessarily mean high levels of competition.


What are the economic disadvantages of a single payer system in the USA? To me, medicare for all seems like a no-brainer economically.

The money individuals, local and state governments pay into insurance premiums, deductibles and medical costs... we simply redirect that into federal taxes that go towards medicare for all.

Plus we give some of that money back, because we no longer have to pay for the income to the insurance company... plus we give back money in administrative costs.

So the country saves money overall with medicare-for-all. What's the downside?

I don't know whether you'll count this as an economic argument, but it's about the role of competition in the allocation of scarce resources, so I think you should:

Suppose that some disease emerges that disproportionately affects the members of some ethnic group, or is concentrated in some geographic area, or is concentrated in some population that is either in or out of political favor with the current administration.

Suppose also that there's some pretty expensive procedure that helps to alleviate symptoms of this disease.

Under a single-payer system, one decision will be made about whether to cover the costs of that procedure.  Quite plausibly, that decision will be either made or influenced by somebody with less than pure motives.  Maybe an out-and-out bigot.  Maybe a politician looking out for the people who  support him.  Et cetera.

I would much prefer that that decision be made independently by multiple competing insurance companies, so that people can choose up front to affiliate with insurance companies that they believe will look out for their interests.

If you think this is all just  speculation, look at what happened when the government took temporary partial control of General Motors.  Politicians immediately started thinking up ways to get General Motors to make decisions that would benefit their own constitutents.

General Motors is one firm, and the takeover was partial and temporary.  Now imagine a takeover of the entire insurance industry, both total and permanent.

You might or might not believe the attendant risks are worth taking --- but you asked what the downside is, and that's a big part of the downside.


Multiple equilibria: which one to select? There are two agents $i=1,2$. Consider the following programm
\begin{align}
&V_1(x_0) := \max_u \int^\infty_0 e^{-\rho t}F_1(x(t),u(t),v(t))dt\\
&V_2(x_0) := \max_v \int^\infty_0 e^{-\rho t}F_2(x(t),u(t),v(t))dt\\
s.t.~&\dot x(t)=f(x(t),u(t),v(t))\\
&x(0) = x_0
\end{align}
where $\rho > 0$ denotes time preference, $V_i(\cdot)$ is the value and $F_i(\cdot)$ an objective function. $x\in X = [0,2]$ is the state variable and $u\in U=[0,1]$ the control of agent 1 and $v\in V=[0,1]$ the control of agent 2 respectively. The state is governed by $f(\cdot)$. The Hamilton-Jacobi-Bellman equation for each agent is given by
\begin{align}
\rho V_1(x)=\max_u [F(x,u,v^*) + V_1'(x)f(x,u,v^*)],\quad \forall t\in[0,\infty)\\
\rho V_2(x)=\max_v [F(x,u^*,v) + V_2'(x)f(x,u^*,v)],\quad \forall t\in[0,\infty)\\
\end{align}

given the respective maximizers
\begin{align}
u^* &= \max_u [F(x,u,v^*) + V_1'(x)f(x,u,v^*)]\\
v^* &=\max_v [F(x,u^*,v) + V_2'(x)f(x,u^*,v)]
\end{align}

such that the HJBs become
\begin{align}
\rho V_1(x)=F(x,u^*,v^*) + V_1'(x)f(x,u^*,v^*)\\
\rho V_2(x)=F(x,u^*,v^*) + V_2'(x)f(x,u^*,v^*)
\end{align}

Symmetric equilibirum

A symmetric equilibirum is given at $\dot x = 0 \Leftrightarrow f(\tilde x,\tilde u,\tilde v) = 0$ with $\tilde x = 1$ and $\tilde u=\tilde v$ and $V_1(\tilde x) = V_2(\tilde x) =: V(\tilde x)$.

Problem

The equilibirum controls $\tilde u$ and $\tilde v$ can't be determinend with the information at hand. The equation
\begin{align}
\rho V(\tilde x)=F(\tilde x,\tilde u, \tilde v) + V'(\tilde x)\underbrace{f(\tilde x,\tilde u, \tilde v)}_{=0}
\end{align}
 is true for every $\{(u,v)\in[0,1]\times[0,1]:u=v\}$. That is, we have multiple equilibria.

Select Equilibrium

My idea is (I made this up, haven't read anything about it) that I select the equilibrium associated with the highest value. We can determine $V(\tilde x)$ for all $\{(u,v)\in[0,1]\times[0,1]:u=v\}$.
Say  $V(\tilde x)$ is monoton increasing in $u$ and $v$, i.e.
\begin{align}
\lim_{u=v\to 0} V(\tilde x) < \lim_{u=v\to 1} V(\tilde x)
\end{align}

Perosnal I would choose the fixed point $(k = 1, u = 1, v = 1)$. I'd like to know if I can motivate it formally as the unique solution.

Do I select the equilibrium associated with the highest value, by definition of the value function?
Can you point me to some literature concerning this point?

Motivating example

Let $F_1(x,u,v) = xu^2$ and $F_2(x,u,v) = (2-x)v^2$ with $f(x,u,v) = v-u$. The HJBs read (with $\rho=1$)
\begin{align}
V_1(x)&=\max_u [xu^2 + V_1'(x)(v^*-u)]\\
V_2(x)&=\max_v [(2-x)v^2 + V_2'(x)(v-u^*)]
\end{align}

Maximizers are
\begin{align}
u^*&= \frac{V'_1(x)}{2x}\in[0,1]\\[2mm]
v^*&=\frac{V'_2(x)}{2(2-x)}\in[0,1]
\end{align}

In symmetric equilibrium we have $\tilde x = 1$ and $\tilde v = \tilde u \Leftrightarrow \dot x = 0$ which gives
\begin{align}
V'_1(1) = -V'_2(1)
\end{align}

The HJb simplifies to
\begin{align}
V_1(1)=\left(\frac{V'_1(1)}{2}\right)^2 = \left(\frac{-V'_2(1)}{2}\right)^2 = V_2(1)
\end{align}

Since both vlaues are equal in equilibirum we proceed with 1. We know from the control space that
\begin{align}
0\leq V'_1(x) \leq 2x
\end{align}

Which is in equilibirum
\begin{align}
0\leq V'_1(1) \leq 2
\end{align}

We can evaluate $V(1)$ for all $V'_1(1)\in[0,2]$ or since $\tilde u=V'_1(1)/2$ for all $\tilde u \in[0,1]$. In the picture I highlighted two possible equilibria $E^A$ and $E^B$. Since the payoff is inreasing with the control we have a higher value associated with equilibirum $E^A$, i.e. $V^A(1) > V^B(1)$.

I'm not sure I follow the logic on that equation having infinitely many solutions and steady states. In any case, in what follows are some guidelines for equilibrium selection.

It depends a lot on the context. Here are some criteria, in descending order of relevance

Clean Ways

Is any of the steady states unstable? If so, it's less likely to be the one that we*re observing in reality. See for example the 0-steady state in the Solow model.
The last point, somewhat generalized; Is there any meaningful starting value? Due to path dependence, you will only converge to one steady state from your meaningful value.
The last point, somewhat generalized; Do we converge more often to one than the other? Basically take a range of meaningful starting values and see whether most of them lead to the same steady state

Believe in Coordination

If all these fail... make the government powerful. Kaplan and Menzio (2015) model unemployment, and can get two steady states given their calibration. They take the lower unemployment level as the "reasonable" steady state, that they use to calibrate parameters against. I don't think they spell it out, but the argument most likely is that if there are two steady states, a powerful government could steer the economy to the steady state implying the highest welfare.

This is basically your "highest value" point - but it should be clear that there is some reason why that is achieved. That's why I said "it depends a lot on the context" - many coordination failure models hinge critically on the assumption that no higher power can steer the economy. See also the Chicken Economy model, a term I believe originally to be coined in Minnesota.

Sunspots

Finally, if all else fails, sun spots are what determine which equilibrium you are in. In economics, we mean by that basically a random exogenous variable, something not modeled. They're very disputed as they have no economic meaning and are hard to digest. Basically, if there was anything that had economic meaning and could determine the equilibrium, you would rather put it into your model than having a coin flip determine it.

See also Karl Shell's summary on the subject for The New Palgrave: A Dictionary of Economics.


When do supply and demand curves shift? Let's assume that the price of apples has risen and that the quantity of apples sold during the last couple of weeks has decreased. From that, we can infer that the supply curve must've shifted to the left.

I still have trouble fully understanding this. I was told that when the endogenous variables, such as price and quantity, change, only a movement on the demand/supply curve happens and when exogenous variables change (demand shocks etc.) the curves themselves will shift to the left or right.

In this example a price increase takes place, so we're just moving up the demand curve, no? And when we reach the target price level, there'll be a corresponding quantity. And now because of market equilibrium the supply curve has to reach that spot, too? So it shifts to the left? Is that correct? So in a demand - supply model, there can't be singular movements on the curves, as there will always be immediate reactions that seek to recreate the equilibrium?

Edit: Thanks for all the answers!

Supply and demand curves are a function of price and quantity.  If anything else changes other than P or Q that is relevant to the curve, the curve shifts.

For supply, these shifters generally fall into three  categories:

Technology
Number of producers
Price of inputs

For demand:

Number of buyers
Price of complements or substitutes
Customer tastes and preferences
Consumer income

If you come up with something that didn't fit into these categories, but is not P or Q, the result is still a shift! You probably just need your imagination to squeeze it into one of these 7 formal categories.  For example, if the supply curve was P=2Q+3, and there was a decrease in the cost of inputs, the demand curve could shift to P=2Q+2.  Note how the price levels are lower at every level of Q.

Changes in supply and demand that are not "shifts" are called "slides along the curve". They are any direct change in P or Q. This is easiest to see with a linear, mathematical example.

Let us say the government wants to set the price of a product. If P=2Q+3 is the supply, then consider that if you set the price (by law) to be 7, then Q is now 2. The curve remains steady, but we slide along it to get to the new P=7, Q=2 position. From wherever P and Q started (P=12 in the example), we now end at point P=7,Q=2, and the curve remains unmoved along the entire line P=2Q+3.


Data Set for Mankiw, Romer, and Weil 1992 I am trying to replicate the results of Mankiw, Romer and Weil 1992, and cannot seem to find the original data.

Have you seen the GitHub Project Replicating Mankiw, Romer and Weil 1992? It seems to have both the data and a replication of the original results.

For the curious, the paper is A Contribution to the Empirics of Economic Growth.
Abstract:

  This paper examines whether the Solow growth model is consistent with
  the international variation in the standard of living. It shows that
  an augmented Solow model that includes accumulation of human as well
  as physical capital provides an excellent description of the
  cross-country data. The paper also examines the implications of the
  Solow model for convergence in standards of living, that is, for
  whether poor countries tend to grow faster than rich countries. The
  evidence indicates that, holding population growth and capital
  accumulation constant, countries converge at about the rate the
  augmented Solow model predicts.


Is there any reason deflation would reduce profit? Profits are the difference between input and output prices. Is there any reason deflation would cause profits to decline generally rather than in those industries where output prices fell faster?

Profits also depend on the quantity demanded/sold. If deflation is accompanied by a decrease in demand, under equal input/output price deflation (or even if input prices fall more than output prices), profits can fall.


Is the satisficing choice function rationalizable if the ordering isn't observable? Edited: Say an observer observes only the choices made by the decision maker (and the sets from which these choices are made), but does not know the ordering. Then would the decision maker's choices still appear to be rationalizable (to the observer)?

Context from Osborne & Rubinstein (2020, Models in Microeconomic Theory):

Yes, of course. That is rationalizable just means that there exists some preference ordering that gives rise to the choices. Observability has nothing to do with it.


Complete markets: security market equlibrium same as contingent commodity equilibrium? "For complete markets, the security market equilibrium is the same as the contingent commodity equilibrium but with far fewer spot markets."

I wrote this down in my notes today on what my professor said but I don't really understand why this is the case. This may be the wrong way of posing a question but I thought this was rather interesting. Any suggestions on the matter are greatly appreciated.

You seem to be looking for information on the Radner equilibrium.

  Radner equilibrium is an economic concept defined by economist Roy Radner in the context of general equilibrium. The concept is an extension of the Arrow–Debreu equilibrium and the base for the first consistent incomplete markets framework.

Further reading:
Jstor: Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets
Handbook of Mathematical Economics: Chapter 20


Exchange rate volatility Colleagues,
I have started an analysis on spatial econometrics models, and looking for data to compute an exchange rate volatility. My variables to construct the weight matrices are all bilateral including FDI, Trade and et.c
My samples are 15 african countries, and is a bit cumbersome to get their bilateral exchange rates. The only possible way is to get rates per one unit of dollar.
* Where can I get a compiled data about it? and/or How could I compute the bilateral counterpart?

It makes more sense to use the USD as an intermediary currency.  If you know the central South African and Nigerian exchanges rates are for example

$14.38\; ZAR = 1\; USD$
$364 \, NGN = 1\, USD$

then by division $\frac{14.38}{364}$ or $\frac{364}{14.38}$

$0.0395\,  ZAR \approx 1 \, NGN$
$25.3 \, NGN \approx 1 \, ZAR$


Are American poverty rates increasing? What about global poverty rates http://www.npc.umich.edu/poverty/

https://www.census.gov/data/tables/time-series/demo/income-poverty/historical-poverty-people.html

I have heard numerous times that the future has gotten better.  Poverty rates are on decline.  Yet, seeing a commercial that 1 in 6 American children are hungry got me thinking.  These sources, and others, have shown poverty in America has gotten much worse over the years.  Due to things like automation, outsourcing, increased taxes due to pay for welfare from the poverty, things will only get worse in terms of these rates.

You should not make predictions in economics.  I understand now.  But, why have we heard so many times things are getting better in these rates and they are not?  More people are hungry.  Places like America, Europe, they are poorer.  Poor may live better, but still not good.

So, my specific question is whether the total fraction of the US population that is poor is getting larger and under what definition of poverty. Thanks.

US

A snapshot of Poverty for the US is here. The key is Figure 5, which shows alternative measures of poverty, using different data and methods. (see text for details). Basically, they show that there is plenty of variation from alternative criteria, ranging from 0% to 5%. This is likely to be an explanation of why sometimes evidence seem to be contradictory.

For the evolution of poverty, see this study. Unfortunately, it stops at 2010, but it is very comprehensive. Figures start in page 53. Basically, poverty has fallen recently, specially when measured in terms of consumption. This could be an indication that before the Great Recession, poorer individuals were living on loans/debt. There is plenty of evidence that consumption inequality is lower than income inequality, because individuals have a desire to "keep up with the Joneses" (the so-called Veblen Effect).

World

Here is the evolution of World Poverty since 1820:

Here is in relative terms (as % of world population):

Here is the current (2014) snapshot of poverty around the world:

An amazing analysis of World Poverty, in simple slides, is here.

A careful analysis of existing evidence is found here.


microeconmics: can a monopolistic firm be profitable if its marginal revenue is negative? Statement: If a monopoly is producing a quantity for which its MR is negative, its total profit cannot be positive.

Is this statement correct or incorrect, and why?

I've read a lot about this, but sources tend to discuss maximizing profitability, while the question above presumes that MR < 0 (and obviously smaller than MC) but wonders about profitability in this scenario.

My hunch is that yes, it is possible, but I'm struggling to prove it.

Edit: this getting downvoted is the reason why SE is deteriorating at the rate it is. I’ll go look for a more welcoming community.

As a counterexample to the statement, consider the following:

Let demand be $p=10-q$, so that $MR=10-2q$. Suppose there’s no cost, so that profit equals revenue. If the monopolist produces at $q=6$, $MR=-2<0$, but $TR=(10-6)6=24>0$, thus contradicting the statement.


Why do Eurozone members have a commonly higher Debt-to-GDP ratio? As you could find out from the map, among 22 countries whose debt-to-gdp ratio is above 50%, only 7 are not Eurozone members, and from which only UK is comparably large economical entity.

Interestingly, according to Euro convergence criteria, to adopt the euro as one country's currency:

  The ratio of gross government debt (measured at its nominal value
  outstanding at the end of the year, and consolidated between and
  within the sectors of general government) relative to GDP at market
  prices, must not exceed 60% at the end of the preceding fiscal year.

So what's the reason behind that most Eurozone members have a higher Debt-to-GDP ratio?

They've already adopted the Euro (from which there's no "easy way out") so the convergence criteria is just dead letter after that...

The EU has a procedure called the EDP (Excessive Deficit Procedure), which seems to apply to both excess deficit (3%) and (in theory, I think) the debt/gdp ratio too (60%). In general it seems the EDP was mainly applied against the former rather than the latter criterion... and even then it took e.g. Spain a decade to come out of EDP (even though they don't meet the latter debt criterion still; it seems the EU just needs to be happy that they have a plan in that regard). The EDP criteria apply to all EU members.

Also, if you're part of a "small currency" you are more easily punished on forex market, as opposed to the bond market where southern Eurozone took their pounding--"Eurozone crisis" and all that.

Based on IMF data, of the non-Eurozone EU members, Hungary and Croatia aren't too far from the Eurozone average and have more debt/gdp than Germany. Poland has less than Germany, but not a whole lot less and they might cross in the next 5 years given the current trends.

In the lower debt/gdp group are two Nordic countries Denmark and Sweden, but also two Eastern countries, Romania and Bulgaria. (The fifth in this group is the Czech Republic.) Romania, Bulgaria and the Czech Republic for example have among the lowest social spending as a ratio of GDP as well. Denmark and Sweden are at the opposite end in that regard; explaining their lower debt/gdp is not that easy. You might want to look at these countries individually...

Denmark apparently had the "crown" in taxes in the EU at in 2019; this could well compensate the higher social spending, as far as government debt goes. For 2018 on which I could find more detailed tax-to-GDP ratios, Denmark and Sweden are very close. Also Denmark and Sweden have experienced growth compared to Germany's, somewhat exceeding the UK's.

The flip side is that there's a high level of household debt in Denmark (130% of GDP in 2018) and Sweden is also well about average(90%) while Germany's (54%) is below EU average (66%)... which in this case is also slightly above Eurozone average (62%) (no doubt the latter is explained by the UK in this case). The Eastern counties are also far below [either] average. (Denmark seldom makes the news in the same sentence as "the IMF", but when it does, it's because of IMF's concern/warning about household debt.)


Perfect Bayesian Equilibrium in a two stage game with incomplete information I would like to solve a game where firms have private information about their own type, but only know the distribution of the other firm's type. They interact in two stages, where the strategies depend on each other. The first period is an optimization problem, and in the second period Nash Equilibria have to be identified.
Example:
Consider two firms, $A,B$ who are trying to sell their product to one customer.
The customer just needs one indivisible unit of the product, which has value V for the customer, and will buy the product from the firm that offers the lowest price.
Both firms initially are able to produce the product at cost $C$, but each firm $i \in \{A,B\}$ may decrease it productions costs by $c_i$ in private by investing $d_i (c_i)^2$.
Each firm $i$ has private knowledge about its own $d_i$, but only knows the distribution of $d_j$. (If the solution needs a specific distribution just pick one which fits your needs.)
The sequence of game stages is:

The two firms simultaneously privately invest into cost reductions $c_A,c_B$.
The firms set prices $p_A,p_B$ for the product.

Straightforward backward induction does not work in my trials, due to the asymmetry of cost reduction investments.
Question: Is there a good textbook or paper where sth. similar is done or do you have any technical knack which I have missed?

In your example, you would still use backward induction to solve for the Perfect Bayesian Equilibrium (assuming the distribution of private costs has full support). In fact, the second stage of your example is similar to a Bertrand competition with asymmetric information. You can refer to the following paper for a general solution of the second stage game:

Spulber (1995) "Bertrand Competition When Rivals' Costs Are Unknown", Journal of Industrial Economics 43:1-11.

The first stage decisions should follow in a straightforward manner.


The Price is Right Bidding Strategies I am currently reading this article.

If you are unfamiliar with the game, "The Price is Right", at the beginning, four bidders are selected who have to guess the price of some consumer object. Whoever gets closest to the retail value (rounded to the dollar) without going over goes on to play a different pricing game. Play is sequential, so you can imagine bidding last provides an advantage. This article is studying whether or not players behave rationally in this first part of the game, or whether it is bounded by some simpler decision rule.

The paper has a proposition:

  Suppose that contestants have rational expectations. Then in equilibrium,

  The fourth bidder must win at least as often as the third bidder; and the third bidder must win at least as often as either the first or second bidder.
  The fourth bidder must win at least 1/3 of the time.
  The first and second bidders together cannot win more than 4/9 of the time.

The appendix then, in its proof, starts the following explanation:

  First note that if a bidder believes that his probability of winning exceeds that of another bidder, then since every bidder has rational expectations, this belief must be correct. Thus to show that a bidder wins more often than another bidder, it is sufficient to show that he believes that he wins more often.

Which is fine.

  First consider the last bidder. Since his optimal strategy is to pick the interval that he believes gives him the largest probability of winning, he must believe that he does at least as well as any previous bidder.

I do not understand this part of the explanation. Is the logic that the last bidder will bid 1 dollar above whoever it will get the best distribution, and then if someone were to have a better chance of winning than the last bidder, the bidder would simply bid 1 dollar above that person?

Yes. Note that in the paper the authors assume that the action space is continuous, so it is possible to bid in arbitrarily small fractions of a dollar.

Suppose players $i\in\{1,2,3\}$ have bids $b_i$ that, without loss of generality, are ordered $b_1<b_2<b_3$. Write $p$ for the actual price.

If we forget about player four for a moment, then players' probability of winning are

$$\Pr(1\text{ wins})=\Pr(b_1\leq p< b_2)\equiv \pi_1$$
$$\Pr(2\text{ wins})=\Pr(b_2\leq p< b_3)\equiv \pi_2$$
$$\Pr(3\text{ wins})=\Pr(b_3\leq p)\equiv \pi_3.$$

Suppose $\pi_1>\pi_2,\pi_3$. Then the optimal thing for 4 to do is to bid $b_4=b_1+\epsilon$ ($\epsilon$ positive and small). This bis would result in
$$\Pr(1\text{ wins})=0$$
$$\Pr(2\text{ wins})=\Pr(b_2\leq p< b_3)$$
$$\Pr(3\text{ wins})=\Pr(b_3< p)$$
$$\Pr(4\text{ wins})=\Pr(b_1\leq p< b_2).$$

One can easilly check that $\pi_1>\pi_2,\pi_3$ implies that $\Pr(4\text{ wins})$ is bigger than any rival winning probability. Likewise, if we had $\pi_2>\pi_1,\pi_3$ then the optimal thing for $4$ to do would be to bid $b_4=b_2+\epsilon$.

edit: The above reasoning works if we take $\pi_1>\pi_2,\pi_3$ (with strict inequality). But what if the inequality is not strict?

Suppose that $b_1<b_2<b_3$, and that $\pi_1=\pi_2>\pi_3,\Pr(p<b_1)$. Then it seems like $b_4=b_1+\epsilon$ must yield a lower winning probability than that enjoyed by player 2 for any positive $\epsilon$. indeed we would have
$$\Pr(b_1+\epsilon\leq p<b_2)<\Pr(b_1\leq p<b_2)=\Pr(b_2\leq p<b_3).$$
Similarly, $b_4=b_2+\epsilon$ will yield a lower winning probability than that enjoyed by player 1 for any positive $\epsilon$.

I think that, in equilibrium, this won't happen so I guess the result is fine. But it makes the proof seem a bit hand-wavey.


During deflation, is there any risk on 'printing' money? Take a specific example: Japan.

In the last several years, the central bank of Japan has eased monetary supply for many trillions of dollars, but the CPI decreased in 2016.

My question: is there any negative effect for Japan to ease so much money?

My logic: 'printing' money have an inflationary effect, so if you print too much money, the CPI will rise high and that is simply bad. However, since the CPI is decreasing even though Japan eased so much money, I wonder there should be no problem for 'printing' even more so long as the CPI does not rise?

But the result of my logic sounds ridiculous. I believe the productivity determines the wealth of a country, but from my logic, if you can 'print' money without inflation, you will be rich too.

You should be confused by Japan's version of QE. No one knows the long term implications, it has never been done before.

I don't know that much about Japan's QE other than the monthly totals, but the concept behind QE is simple.

Normally, when a central bank wants to stimulate economic activity, they will do so by reducing the returns on saving and therefore incentivize consumption. To do this, they need to lower market interest rates.

In the past, they usually did so by controlling the rate at which banks lend to each other, or to the central bank. In the US, this is the Federal Funds Rate. But what happens when you have reached the zero lower bound of the central bank's rate and you need to further influence market rates. In the case of the US and Japan, the answer, in part, was to have the central bank buy debt. But it was not just a small amount like they normally would to change their rate, QE saw the central banks buying huge amounts of debt.

This debt purchasing was designed to serve a singular purpose: drive down long term interest rates. Why would this be helpful? Well with low short term rates, the market for interbank lending would be very liquid, and would not risk the so called "crunch" of credit that occurred during the recession of 07-08. But that was done almost immediately as the crisis revealed itself.

When QE kicked in in earnest, long term rates did begin to go down, but it did not have the immediate effect that central banks thought it would. Unlike short term lending, the long term market is much more slow to adjust to changes in prices, and to some extent the Fed got around that issue in the US by providing forward guidance on their plan to continue QE "For an extended period of time". Even now in the US, QE rolls on, but under the guise of reinvesting the proceeds of original QE assets into even longer term securities.

But Japan started QE earlier! They practically invented it! Why didn't they recover as quickly as the US?

Well they kind of did. In 2010 they posted 4.2%gdp growth after -5.4 the year before. Unfortunately they didn't maintain it.

The reality is that most highly developed countries, (US, Germany, Japan, ect.) are in a period of stable but slow growth. In fact this is the norm in history. It was only the massive upheavals of the 20th century that allowed for massive GDP growth in developed countries. I think it is unreasonable to expect an average of 4% GDP growth in developed countries in the current epoch.

But we've also seen a positive side effect of more sedate GDP growth: lower inflation. There are plenty of theories as to why inflation is lower in developed countries than it has ever traditionally been. I like parts of all of them. One is that globalization and fewer barriers to international trade has lowered prices of goods and services globally by increasing efficiency.

Another theory is that the strategies of central banks are more effective at maintaining low inflation than they are at bringing down high inflation. In particular, inflation targeting regimes, such as the one currently employed by the Fed, seem to be effective at maintaining well anchored market expectations, and help to reduce market shocks.

As to whether a country can become richer by printing money without inflation the answer is: kind of...

There is interest rate risk to QE. If rates go up, QE loses money for the central bank. But since the central bank controls interest rates, they don't really have to worry about it. The Fed will make money on QE, and the BOJ will probably make money on their QE (although they bought non-government assets in addition to government, so I'm not as sure about those). Does making money on QE make the country richer? Not really. But does lowering the yields on government bonds make a country relatively richer? Yes. With cheaper government debt, the government is more able to conduct expansionary fiscal policy without having to worry about too much interest. In the case of QE it is even better because your own central bank owns the securities so you practically are performing seigniorage!

Of course this assumes that the government then uses this money to improve the production of the country, but if you throw enough money at it, it can't help but make your country richer.

Now, the final question is: what happens in the long term as a result of QE? Is there a risk of inflation as QE securities mature?

I've thought about that question a lot in the last few years, and I think that in the case of the US there is limited risk of inflation from maturing securities. The fed can roll them off at a very slow rate, and by tapering the reinvestment of the securities, they can ensure a slow, gradual decrease in the size of their balance sheet, and low inflation.

In Japan the issue of the types of securities is more of an issue. The US only did QE through government backed securities and mortgage backed securities (to a lesser extent) and both are flexible with their maturity schedules.

I have not looked at the maturity schedules of the BOJ's QE securities, but I hope that they have the same flexibility to reinvest at least some of the proceeds of maturing securities.

Regardless, monetary policy has certainly entered a new era of complexity, and there are risks to the new strategies in terms of accurately communicating the intentions of the central bank in a timely manner. The biggest risk of inflation isn't anything financial, it is the risk of the wrong signals being sent to the markets of the world by the communications of central banks.

I'll try to add citations at a later date (I have several which I was thinking of as I wrote this answer).


Bayesian-Nash equilibrium in a first-price auction In a famous textbook example of a Bayesian-Nash equilibrium, there is a first-price auction with two independent players. Each player $i$ values the item as $v_i$, which is  distributed uniformly in $[0,1]$. It is assumed that the strategy of each player $i$ is to bid a fraction of his value, i.e:

$$ b_i(v_i) := a_i \cdot v_i$$

for some constant $a_i$. Then, we can conclude that, for every $a_1$, the best response of player 2 is to pick $a_2=1/2$. Hence, the only Bayes-Nash equilibrium of this form is $a_1=1/2, a_2=1/2$.

Now, I tried to slightly generalize the example by assuming the values of the players are distributed uniformly in $[c,d]$, for some constants $d>c>0$. But I did not find an equilibrium. Suppose player 1 bids $ a_i \cdot v_i$ and player 2 bids $ $a_2 \cdot v_2$. Then the expected utility of player 2 is:

$$ProbOfWinning(a_2)\cdot GainWhenWinning(a_2)$$

$$=Pr[a_1 v_1 < a_2 v_2]\cdot (1-a_2)v_2$$

$$=\frac{(a_2 v_2 /a_1)-c}{d-c}\cdot (1-a_2)v_2$$

Taking the derivative of this expression w.r.t $a_2$ gives that the best response of player 2 is:

$$
a_2 = {1\over 2} + {a_1 c\over 2 v_2}
$$

so the best $a_2$ depends on $v_2$. This means that the function from value to action is not linear as I initially assumed.

Do I have any mistake in the calculation? What is a Bayes-Nash equilibrium in this auction?

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0.

A somewhat intuitive reasoning is that if the valuations are uniformly distributed over $[c,d]$ and $b_i(v_i)$ is a symmetric equilibrium, then $b_i(v_i)+k$ should define a symmetric equilibrium when valuations are uniform over $[c+k,d+k]$ as neither the probability of winning nor the gain changes.

Assuming a strategy of the form $\alpha_i+\beta_i \cdot v_i$ you get

$$
=Pr[\alpha_1 + \beta_1 \cdot v_1 < \alpha_2 + \beta_2 \cdot v_2] \cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

$$
=Pr\left[v_1 < \frac{\alpha_2 - \alpha_1 + \beta_2 \cdot v_2}{\beta_1} \right] \cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

$$
=\frac{\frac{\alpha_2 - \alpha_1 + \beta_2 \cdot v_2}{\beta_1}-c}{d-c}\cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

Taking derivative w.r.t. $\alpha_2$ and $\beta_2$ you get your first order conditions. Assuming symmetry you get a unique solution. If $c = 0$ then $\alpha_1 = \alpha_2 = 0$.

By the way, I prefer treating $b_2(v_2)$ as the decision variable, then you can
$$
\max_{b_2(v_2)}
Pr\left[v_1 < \frac{b_2(v_2) - \alpha_1}{\beta_1} \right] \cdot (v_2 - b_2(v_2))
$$
and the last fraction is somewhat simpler.


Can a firm have a negative turnover during a period of time? Can a firm have a negative turnover during a period of time? I'm asking this because I couldn't find any details on the web.

I am working with a data table that holds a lot of firms and their turnovers for each month. Some of them have negative turnover for one or more of the months.

Is that possible? How does a negative turnover work? How does it appear in the first place?

One possible circumstance relates to firms undertaking long term contracts (eg for building or engineering work). Where a contract is uncompleted at the end of a period, the firm may (depending on its accounting policies) include in its turnover an amount reflecting its assessment of the work in progress.  Sometimes unforeseen difficulties in completing a contract will subsequently come to light, and it may become necessary to include an offsetting reduction within turnover for the next period. If the contract is a large proportion of the firm's activity, then this could result in overall turnover for that period being negative.


Value of Currency Itself What is the value of an individual piece of currency itself based on the efficiencies it allows in trade, in contrast to the value it represents?

For example, people often talk (complain) about the cost of manufacture of a single US cent coin being greater than one cent — but to me, this seems an irrelevant comparison.  A single coin will be used in many, many transactions over its lifetime, facilitating the efficient transfer of (possibly) thousands — or tens of thousands — of dollars.  Surely that is worth far more than one cent.

Without use of that coin, the parties involved may have had to round prices up/down, purchase multiple items at once, or forego certain transactions altogether.  Smaller denominations are useful by the same logic that drives public companies to split their shares, making it possible for people to purchase just a single piece of "penny candy" if they would like, as opposed to being forced to buy "five for a nickel" (or none at all.)

Beyond that, physical currency has a value simply in facilitating efficient trade.  If there weren't enough coins to go around, merchants may have to wait for somebody else to spend them (or the government may have to mint more, which raises its own issues since that also involves introduction of additional inflationary value) before customers have enough to conduct trades.

I realize these are somewhat unrealistic scenarios, since in practice there are typically enough pieces of currency around that trade isn't significantly impacted.  But this shouldn't change the fact that the currency is still increasing the efficiency of trade, in however minor a way, and that in virtue of the large number of trades in which those pieces of currency will (likely) be involved, the aggregated value becomes quite significant.

I want to strongly emphasize here that I am not looking for answers that address things like forex value, etc.  I've found it extremely difficult to even research this topic because of the multiple meanings of all the terms involved (just try searching for "use value of currency" and you'll see what I mean) and a satisfactory answer need only point me in the direction of appropriate articles/books, so long as they directly address the aspect of currency in which I'm interested here.

EDIT: I should also clarify that I'm not primary interested in coins.  That's just an example.  I'm interested in how to estimate the value that a currency provides through facilitating financial transactions, regardless of the physical nature of the currency (e.g. paper, metal, whatever.)

The value of something, in this case money, is based on it's demand; and since you want the value based on it's efficiency, not what it can purchase, you are looking for the value of money based on how efficient money is.
I believe I have just the answer for you.

Your missing variable is the Velocity of Money.
The demand for money and the Velocity of Money are inversely related. (your question emphasizes this point, "in contrast to the value it represents?".

The velocity of money (or the velocity of circulation of money") is a measure of how fast money passes from one holder to the next. It is most commonly measured as the income velocity of money, which is the frequency at which the average unit of currency is used to purchase newly domestically-produced goods and services within a given time period. In other words, it is the number of times one unit of money is spent to buy goods and services per unit of time.

Vt = PT/M
where

Vt is the velocity of money for all transactions in a given time frame;
P is the price level;
T is the aggregate real value of transactions in a given time frame; and
M is the total nominal amount of money in circulation on average in the economy.
Thus PT is the total nominal amount of transactions per period.

Hope this helps.


I lost $100 in the laundry, who won? I lost a 100 dollars bill in the laundry (it was destroyed).
I clearly lost 100 dollars.
Where are they now? Who won 100 dollars?

The central bank that issues the currency won; they can now issue an additional $100 without increasing the price level. This is similar to seignorage rents due to overseas circulation— in fact, to a central bank, the two are indistinguishable, as the central bank would be unaware that your note had been destroyed.


How to determine the cost of a good Can the cost of any good be determined based on its ratio to a minimum wage?

For instance, if 10 years ago, eggs cost \$2 / dozen and the minimum wage is \$10 / hr (eggs cost 20% of the minimum wage) and at present, due to inflation, eggs cost \$2.50 / dozen and the minimum wage is \$15 / hr (eggs now cost 16% of the minimum wage).

Could it be said that the cost of eggs have decreased despite a rise in prices due to inflation?

While you're on the right track, minimum wage is not the best way to get at it. What you're discussing is called "real wages", and it is usually calculated based on some average wage, as many workers make more than minimum wage. So yes, the price of eggs has gone down (if minimum wage is a useful metric for average income).


Why is GDP equal to the sum of revenues? National Income can be expressed in two ways :

In can be derived from the GDP, as, roughly, a sum of "money spent" : GDP is equal to C + G + I + NX, where C, G, I and NX are houshold consumption, government expenditures, investment, and net exports. National Income is then obtained by adding net income from abroad and removing consumption of fixed capital.
It can be calculated as the sum of incomes (details here) : mainly pre-tax employee compensation, pre-tax corporate profits, and taxes on production and imports.

The fact that the two ways give the same results (let alone the "statistical discrepancy") is usually taken for granted.

Of course the principle makes sense, but I am looking for a rigorous approach / mathematical proof.

One could say that whatever agents earn, they spend. But what if they spare some money ?
One could say that whatever is spent, is earned by some agent. But I tried this and couldn't get rid of the 'I' in the first formula.

Could somebody enlighten me ?

It is right that the whole savings are invested.

You can use the (simplified) national income and the national output to show the equality.

The national income is $NI=S+I$

The national product is equal to the national income: $NP=NI$

The national product is equal to the sum of consumption and investment: $NP=C+I$

By using this three equations it can be shown, that $I=S$.

This presumption is made for the IS-Curve at the IS-LM-Model.


What would be the effects of an expiration date on currency? Let's say we introduce a monetary system where each piece of money is printed with an expiration date - a year, perhaps.

I'm imagining something rather different from inflation (although I do not know whether it would actually behave much differently) :
With inflation, the entire money supply gradually devalues at some set rate.
With an expiration date, there is no devaluation for a while, and then a sudden drop. However, at that point (and in fact for quite some while already) there will be new money in print, which will still have the same value the old money would have had a day ago (plus/minus the undoubtedly still existing inflation rates on top of that, but unless that's somehow important in this, let's ignore that for the purpose of this question.)

So the setup would essentially be a government infinitely printing money BUT to prevent hyperinflation, old money is suddenly and permanently lost from the economy. Presumably the printing rate would roughly coincide with the expiration date such that, at any given, about the same amount of money (plus/minus external demand but let's ignore that as well) is available. It may just suddenly change owner.

What will happen, short- and long term? (Obviously short time a lot of things, like savings plans and how banks work would have to significantly change or they would downright fall flat, but after a while, assuming our hypothetical government wouldn't just frightenedly revert the laws after two years of resulting economic failure or something, a new economy would build around these new restrictions. What would that be like - without delving too deeply into pure speculation. Are there useful models adaptable/built for such a situation?)

A bill with an expiration date does not become worthless suddenly. Clearly its value would decrease over time until, just before the expiration date, it was next to worthless. Think about it: how much would you pay for a $1 bill that expires in 5 minutes? Certainly a lot less than you would for one that expires in 6 months.

This constantly changing value would make it hard to use as a medium of exchange. Whenever buying or selling something, one would have to look not just at the face value of the bill, but also at the expiration date, to be able to calculate its real value. People would probably start carrying calculators next to their wallets.

Cattle (which obviously "expire") were once used as currency, so it certainly could work. But that does not mean it is ideal.


Is the fundamental analysis also a self-fulfilling prophecy? John J. Murphy in the first chapter of his book "TECHNICAL ANALYSIS OF FINANCIAL MARKETS" says that concerns regarding fundamental analysis being a self-fulfilling prophecy are seldomly raised, as opposed to technical analysis which receives criticism in the form of getting labelled as a self-fulfilling prophecy. So is the fundamental analysis, at-least in a few ways,  a self-fulfilling prophecy?

If a sufficient fraction of market participants follow technical analysis it can work as self-fulfilling prophecy. If some pattern occurs that technical analysis interprets as signalling an imminent drop in prices these people will sell, which will cause the prices to actually drop.
As is obvious from the above argument, whether an interpretation of the data can become a self-fulfilling prophecy, depends not on the nature of the the interpretation, but on whether a sufficient amount of market participants are prepared to act upon it. For example if fundamental analysis shows that a firm is likely to go bankrupt other firms become hesitant to trade with it because if it fails without settling its bills with them they will have to take the losses. This in turn makes it harder for the threatened firm to survive.


Microeconomics theory and integration by parts (proof). I understand this part: $\int x^\prime (z)F(z) dz+\int x(z)f(z)dz=\int zf(z)dz \rightarrow \int \frac {dx}{dz} F(z) dz+\int x(z)\frac {dF(z)}{dz} dz=\int zf(z)dz \rightarrow x(z)F(z)= \int_{0}^z tf(t)dt$

Then, the author says obtain the following solution by integration by parts.
$x(z)=z-\frac {\int_0^zF(t)dt}{F(z)}$

I don't know which term should I integrate so that I get to that result.

Thank you in advance.

Integrate the $f(t)$ (a primitive of which is $F(t)$) and differentiate the $t$. This yields

\begin{align*}
x(z) F(z) & = \int_{0}^{z}{t f(t)dt} \\
& = \big[ t F(t) \big ]_{t=0}^{t=z}-\int_{0}^{z}{F(t)dt} \\
& = z F(z)-\int_{0}^{z}{F(t)dt}.
\end{align*}

Dividing by $F(z)$ on both sides yields the result.


Calculating Weighted Average Cost of Capital (WACC) in this example I am stuck with an exercise where I have to calculate the Weighted Average Cost of Capital (WACC) of the company X.
The data is as follows:

We have two periods (t = 0 and t = 1). Company can be found in t = 1 in two states. Each state has a probability of 0.5. In t=1 the value of the company in State 1 is 200.000 USD and the value of the company in State 2 is 70.000 USD.
The corporate tax rate is 40 %. The incurred costs if the firms becomes insolvent would be 20.000 USD in t = 1, which is subtracted from the firm's value. Contingent unit claims are both in State 1 and State 2 equal to 0.45 USD.

Since I am not the economist, I would have the following questions. In order to calculate WACC we need to calculate cost of equity (Re) and cost of debt (Rd). I found in the literature that the cost of equity (Re) is obtained by CAPM (capital asset pricing model), where Cost of equity (Re)  = risk free rate - beta * (market risk rate - risk free rate)
Now let us suppose that the firm's capital is 100% equity financed. I have no idea of how to calculate the cost of equity using CAPM model. Using the data that I have at disposal I guess that I can calculate risk free rate by inversing the sum of contingent claims in State 1 and State 2 which is equal to $\frac{1}{0.90}$
The problem remains how to calculate $beta$ and market risk rate having at disposal only data that I presented above.

You can't. Beta is observed from the value of the company respective to the market (beta represents volatility).
Also the market risk premium is the expected return for the market portfolio and cannot be calculated with what you have been given.


Spurious regressions If $\{ X_t \}$ is a $I(1)$ series and $\{ Y_t \}$ is a $I(0)$ series, would it cause "spurious regression" when regressing like $Y_i=\beta_0+\beta_1 X_i+u_i$? Thanks!

That is illegal because an I(1) series wanders ( doesn't have a constant mean )  and an I(0) series doesn't so they can't be set equal. In order to obtain a valid time series regression, the order on the LHS has to be the same as the order on the RHS.

Note that if both sides are I(1), then a time series regression is okay but only if what is on the RHS is cointegrated with what is on the left. If they aren't cointegrated and both are I(1), then that's what is called a spurious regression. ( i.e: a non-sensical regression ).


How to determine covariant stationary values? I am trying to determine the values for when this ARMA model is covariance stationary.

I have the model: $z_t = a + Bz_{t-1} + u_t + u_{t-1}$

I have written it in terms of the lag operator:

(1 - BL) $z_t= a + (1 + L)u_t$

However I am unsure what to do about the constant?

Hi: Here's an attempt at a heuristic solution.
Dividing both sides by $(1 - \rho L)z_{t}$ ( using $\rho$ instead of B and $\epsilon_t$ instead of $u_{t}$ because that notation is easier for me ),
gives
$z_{t} = \frac{a}{1-\rho L} +  \frac{\epsilon_t}{1 - \rho L} + \frac{\epsilon_{t-1}}{ 1 - \rho L} $
= $\frac{a}{1-\rho L}  + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i-1}$
This means that we have a similiar expression for $z_{t+1}$.
$ z_{t+1} = \frac{a}{1-\rho L}  + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i+1} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}$
To calculate $cov(z_{t}, z_{t+1})$, the first term on the RHS doesn't play a role since it's not random.  Now,  since the $\epsilon_{t}$ are independent, one only needs to find the terms in the two expressions where the same $\epsilon_{t}$ occurs because that will result in a non-zero covariance term. The second infinite sum in $z_{t+1}$ coincides exactly with the first infinite sum in $z_{t}$ so all of those terms coincide without any shifting.  Next,  the terms in the second infinite sum of $z_{t}$, shifted over two time periods, coincide exactly with the terms in the first infinite sum of $z_{t+1}$ so all of those terms coincide.
So, now it should be clear that $cov(y_{t}, y_{t+1})$ will be equal to $cov(y_{t+k}, y_{t+k+1})$ because, with the terms in each expression be shifted  over by k time units, the term terms that coincide won't change. Of course, this is a heuristic argument but hopefully you can work out the details.


Linear Homothetic Utility A Homothetic Utility is where
$$
\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax,ay)=au(x,y)
$$ (or its monotonic transformation).
A linear Homothetic utility is defined as $$
\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax+b,ay+c)=au(x+b,y+c)
$$
where $b,c$ are constants.
This preference has very similar property as the homothetic preference. In fact, if we simply translate the coordinate system in the direction of (b,c), then the preference becomes homothetic.
Are there any works covering this property? I've checked a lot of theory papers in homothetic preference but found no luck.

Homothetic Preferences by James DOW· and Sergio Ribeiro da Costa
WERLANG
Homothetic and weakly homothetic preferences by J.C. Candeal, E.
Indurain
Linear-homothetic  preferences, by B Datta, H Dixon

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the utility function that leads to the Linear expenditure system.
To see that it is linear homothetic notice that:
$$
\begin{align*}
u(\beta \tilde x + a, \beta \tilde y + b) &= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\
&=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\
&=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\
&= \beta u(\tilde x + a, \tilde y + b).
\end{align*}
$$
It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.
For example you could define a CES utility function with subsistence levels:
$$
u(x,y) = (\alpha_x(x -a)^\sigma + \alpha_y(y-b)^\sigma)^{1/\sigma}
$$
This will also satisfy linear homogeneity. This paper of Baumgärtner, Drupp & Quaas does something like this.
In general, if you take any homothetic utility function $u(x,y)$ then the modified 'subsistence-augmented' function:
$$
\tilde u(x, y) \equiv u(x - a, y - b),
$$
will be linear homothetic.


Which single letter abbreviations are typically used for the production factors Land, Labour and Capital? I often see "L" for Labour and "K" for Capital, as well as "M" for natural resources (i.e. Land) but they don't seem to be applied consistently.

In economic models, K is commonly used to represent "capital." This is presumably due to the fact that German for capital is "kapital," and also to the fact the C is more commonly used to represent consumption.

There is no difference in meaning between labor and labour. Labor is the preferred spelling in American English, and labour is preferred throughout the rest of the English-speaking world.

I believe it really depends what country the economic authors or book writers are from.


Why is the income effect zero for quasilinear utility functions? Suppose I have the utility function $$U(x,y) = \sqrt{x} + y$$
subject to budget constraint
$$p_x x + p_y y = m$$
Then
$$x_M =\frac{p_y^2}{4 p_x^2}$$
$$y_M = \frac{m}{p_y} - \frac{p_y}{4 p_x}$$

where $M$ denotes Marshallian.

Now suppose I increase $p_x$ to $p_x'$.

Why is the income effect zero?

From the formula for $x_M$, we see it has no dependence on income $m$. So $$\frac{\partial x_M}{\partial m} =0$$ Thus, the Slutsky equation

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +-\frac{\partial x_M}{\partial m}x_M$$

implies

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +(0)x_M $$
$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x}  $$
Hence, and since $-\frac{\partial x_M}{\partial m}x_M$ is the income effect, this implies the income effect is zero and all the change is due to the substitution effect.


Is quantitative easing part of the ECB's and Fed's mandates? Is quantitative easing part of the ECB's and Fed's mandates or is it an exceptional policy that they used, especially in the financial crisis 2007-2008?

It’s a means to an end, and to the mandate that these institutions imposed on themselves. We could also call it an “unofficial” mandate since politicians would likely put pressure on them in indirect ways to uphold their primary mandate. That primary mandate is a 2% inflation target. QE of course expands the money supply which can expand the velocity of money and keep prices relatively stable, as we already know. So QE is a means to that mandate end so that prices don’t fall sharply as we were beginning to see following the financial crisis. QE in it of itself is not a mandate.


What do they mean by “bottom line” and why did US IRS turned back 810 millions to IBM? and what do they mean by “unspecified “ layoffs? In the quarter ended in March, IBM’s revenues rose by 5.1 percent to \$19.1 billion, and its gross profits were up by 6.1 percent to \$8.25 billion. There were a whole bunch of things that affected the bottom line in the quarter, including a settlement of two audits by the US Internal Revenue Service that gave it back \$810 million, but IBM then took \$610 million in restructuring charges for unspecified layoffs. The company also took some charges – IBM’s chief financial officer, Jim Cavanaugh, did not say how much – that pushed its overall systems business into the red ink.

Why did US IRS gave 810 to IBM? and what do they mean by saying bottom line?

As regards your second question,
"Bottom line" is a descriptive term for profits/losses, visualizing that when we calculate whether we had profit or loss,  we start writing revenues then expenses and at the end we have a line that tells us the difference between the two. But since it is the "last line" it is also at the "bottom" of what we have written, hence "bottom line".

Sometimes it is also used figuratively to indicate the "end result" of something other than financial activity, especially with an intent to brush away reasons and excuses why something did not happen or happened inadequately.


Impulse Response Functions for non-linear models? The notes I'm reading define the IRF as $\text{IRF}(h)=E_t(X_{t+h})-E_{t-1}(X_{t+h}|\text{shock}_t)$. All the examples they show are linear, i.e. ARMA. However, they say that for nonlinear the formula gets harder. But they don't show explicitly how to do it. So,how does one compute the IRF for non-linear models?

Any help would be appreciated.