Inclusion of firms on the efficient frontier of a Production Possibility Set I'm working through Cooper's Data Envelopment Analysis, and they are going over Production Possibility Sets.

They present 9 firms, each with two inputs and one output:

Graphically it easy to see why firms (E,D,C) make up the efficient frontier, however I am struggling to find a mathematical motivation for the inclusion of firm D? How do you show its input/output ratios warrant its inclusion algebraically as opposed to graphically.

Though I echo many of the sentiments posed by @AlecosPapadopoulos, maybe this would be helpful for your understanding. Ultimately, we can consider the efficiency frontier as the firms that make up the most efficient ways (or methods of production) of generating a particular output. For example, if our initial bundle of inputs was (listing them in $(x_1, x_2)$ pairs) $(2,5)$, the only firm that can generate a unit of output is firm E, hence its inclusion on the efficiency frontier.

But what about any bundle like $(4,3)$ or $(5,2)$? Clearly firms A and D can produce one unit of output with the first bundle, and firms D and F can generate one unit of output with the second bundle. However, in both cases, only firm D is the most efficient producer.

(Why do we care? Perhaps if we're trying to consider the firm that satisfies a cost minimization problem. Suppose the price of each unit of $x_1$ is $1$ and the price of each unit of $x_2$ is $2$. Which firm will produce with the lowest cost? Firm D's production costs are only $\$8 $ per unit of $y$ (while both C and E's costs are $\$10$ per unit of output!), and hence if all goods are undifferentiated the cost curves are constant, firm D will dominate the market).

Hope that clarifies a bit- let me know if I totally misunderstood your confusion.


In microeconomics : is this the contradiction in the atomicity of firms ? Let $p$ be the market demand. It is a function of the market production $Q$. Let $q_i$ be the production of firm $i$.

Reading Steve Keen (in Debunking Economics, chapter II) quoting George Stigler, I think the first wants to deduce the following contradiction in the atomicity of firms.

Using the chain rule, we get : $\frac{dp}{dq_i}= \frac{dp}{dQ} \frac{dQ}{dq_i}$.

The firm $i$ is price-taker, so the market price is the same whatever its production and so $\frac{dp}{dq_i}=0$.

The demand $p$ is a (strictly) decreasing function of $Q$ (supposing the law of demand is true). Thus $\frac{dp}{dQ} < 0$

The other firms than firm $i$ are not supposed to react to a change in production of firm $i$, so that $\frac{dQ}{dq_i}=1$.

We get : $0 < 0$. Is that the contradiction that Steve Keen means (or another way of expressing it) ?

Thank you so very much !

A price-taking firm takes prices as given, but that does not mean that the firm cannot influence prices; it just means that the firm ignores its own impact on prices.

Now the question is how sensible it is to assume that firms take prices as given. The usual view is that it is a reasonable assumption when the impact of a firm on prices is small enough that profit-maximizing behavior for given prices does not differ much from profit-maximization under actual prices. This is usually the case if there are many firms that are small relative to the market. There are various ways to make this precise. One can also work with models with a continuum of firms in which a single firm hs literally no impact on prices.

As a side remark, Steve Keen's book reveals a strange mixture of incompetence and dishonesty. Keen is ignorant of basic calculus. Keen, sometimes with coauthors, has come up with a nonsensical theory of the firm. Here is a note by Paulin Anglin pointing out several of the many problems. One of these problems is acknowledged here. Later, Keen (and Standish) wrote a survey of Keen's history of criticising the theory of the firm in which they did not acknowledge Anglin's criticism. I recommend reading books from people with higher scholarly standards than Keen.


Why some companies are reporting loss due to the new US tax law? From http://uk.businessinsider.com/bank-of-america-fourth-quarter-earnings-q4-2018-1?r=US&IR=T:

  Like the other big banks, Bank of America took a hit from the recently
  enacted tax law, writing down \$2.9 billion primarily in deferred tax
  assets that declined in value.

  JPMorgan earlier reported a \$2.4 billion fourth-quarter loss because
  of the new tax law, and Citigroup reported a $22 billion loss.

My simple minded expectation is that when you lower a tax rate companies should have more money.

I tried reading about 'deferred tax assets':
https://www.investopedia.com/terms/d/deferredtaxasset.asp

Where I found:

  The simplest example of a deferred tax asset is the carry-over of
  losses. If a business incurs a loss in a financial year, it usually is
  entitled to use that loss in order to lower its taxable income in
  following years. In that sense, the loss is an asset.

So perhaps there is a place for clever accounting around expenses and taxes. But what is the exact mechanism? Why some companies are reporting loss due to the new US tax law?

The lower tax rate means that they will gain less benefit when they apply their deferred tax assets against their income.

They have large amounts of these assets recorded on their balance sheets and are adjusting them to reflect the new, lesser value.


How to solve Bertrand Equilibrium with a non-constant MC? I know that Bertrand Oligopolies will charge a price equal to marginal cost. But if marginal cost is, say,

  2Q or 4Q^2

(i.e. not constant) then how does one determine where, on the MC curve, the equilibrium lies?

If there are 5 Bertrand firms, then do I just find where MC intersects the Demand Curve and divide the quantity by 5, and then set price equal to MC at that quantity?

Conceptually this sounds reasonable, but I'm not sure that this is correct. If these firms charge this price then it seems to me that one of them will under produce to charge a lower price until (if MC is 2Q for instance) price and quantity = 0. So that no firms will produce because if they do then someone will charge a cent less till we reach 0 again.

The main issue seems to be that you assume that under Bertrand competition a firm is free to set a price and a quantity as well. But under Bertrand competition firms set prices and then have to meet the demand whatever level it takes. Below is a detailed discussion.

I will assume that cost functions are the same for all firms and also that the marginal cost function is non-decreasing. (You do not explicitly state these conditions but seem to assume them as well.) I will also assume that demand is non-increasing in price.

In this case, your candidate for the equilibrium is when all firms set the price $p^*$, where all firms produce
$$
q^* = \frac{D(p^*)}{5}
$$
and where
$$
p^* = MC(q^*).
$$
Is this really an equilibrium, or could a firm profit by deviating to a slightly lower price $p'$?

If say firm 1 were to deviate to $p' < p^*$ while the other firms still charge $p^*$ then all consumers would seek out firm 1. Since $p' < p^*$ and demand is non-increasing in price
$$
D(p') > D(p^*).
$$
Firm 1 has to fulfill this alone, so
$$
q_1 = D(p') > D(p^*) = 5 \cdot q^*.
$$
Because marginal cost is non-decreasing you also have
$$
MC(q_1) > MC(q^*).
$$
So now firm one charges a price lower than $p^* = MC(q^*)$ but it has to produce so much that the marginal cost is larger than $p^*$, hence it earns negative profits, hence deviation from $p^*$ was not profitable.


Why is the velocity of money in an economy affected by the frequency in which the worker is paid (like weekly, bi-weekly, monthly, etc.)? I don't have any economics background, and was going though a course on Macroeconomics. I was solving the practice question here: https://mru.org/practice-questions/causes-inflation-practice-questions?sid=2719557

The last question says:

The velocity of money is affected by which of the following? *

a. How quickly the treasury prints new money.

b. If a consumer makes purchases with large bills or smaller bills.

c. If workers are paid weekly, bi-weekly, or monthly.

d. If a consumer uses a payment plan to purchase something or pays outright.

The answer is c. I cannot understand why does the frequency of payment matters. Also, if c is correct, why is option d wrong?

This is because by definition velocity of money is basically how quickly people spend money they get. Hence any factor that causes people to hold money holder decreases velocity of money and vice versa.

The frequency people get pay might affect how quickly they spend money. For example, if you are paid quarterly basis you will probably be more conservative with your spending to make sure your money lasts for whole quarter (extreme example just to show the point) then if you are paid on monthly basis and you will be even less conservative with your spending if you get paid every week or every day.


How to interpret increases in percentage with different sample groups Apologies if this question is not worded very well - this is not really my field of expertise but I'd like to see how to correctly understand some data.

Let's say I am studying the increase in cases of a disease over the last 100 years. I am comparing two different ethnicities, to see if one of them is more susceptible to the disease. The data I have is as follows; for the purposes of the example let's assume that all other factors are equal:

1917: General population consists of 70% Ethnicity A, 30% Ethnicity B

1917: 10% of Ethnicity A contracts the disease during their lifetime

1917: 30% of Ethnicity B contracts the disease during their lifetime

2017: General population consists of 60% Ethnicity A, 40% Ethnicity B

2017: 30% of Ethnicity A contracts the disease during their lifetime

2017: 60% of Ethnicity B contracts the disease during their lifetime

I can make two observations:

The percentage of infection has doubled in Ethnicity B, and tripled in Ethnicity A.
There is a 30 percentage point increase in Ethnicity B, but only a 10 percentage point increase in Ethnicity A

To someone who has a limited grasp of statistics and data analysis (i.e. me), these two outcomes seem contradictory - observation 1 indicates that Ethnicity A is worse off than Ethnicity B with regard to the rate of increase, while observation 2 seems to indicate the reverse.

From a data analysis and statistics point of view, what is the correct approach to take to determine the significance of the respective increase in percentages?

Also, does the changing ratio of ethnicities in the general population also affect how I should interpret the results?

If you ask me your question is better suited to cross validated (stats.stackexchange.com).

Anyway you're probably confused because the percentages refer to different base lines. Note that the 60% of disease at B in 2017 refers to 40% of the population, whereas the 30% in 1917 refers to 30% of the population, so they are hard to compare.

Try working it out in absolute numbers to get a clearer picture:

In 1917 if there are 100 people, 70 of ethnicity A, 30 of B. Of the 70 people of A 7 (10% of 70) get sick, of B 9 get sick (30% of 30)

In 2017 there are 100 people, 60 of ethnicity A, 40 of B. Of the 60 people of A now 18 catch the disease (30% of 60), and of B 24 catch the disease (60% of 40).

Observations: overall sickness has increased (16 vs 42). The absolute increase in sick B people (15) is larger than that in A people (11), but this in part due to the fact that there are more B people now


Properties of Financial Markets in Real Life When studying financial economics, three concepts appear everywhere

Equilibria (investors maximise utility, markets clear and aggregated demand equals aggregated supply)
Completeness (there are enough (linear independent) assets to match the number of states of nature, i.e. payoffs can be hedged/replicated)
Absence of arbitrage (there is no free lunch, no self-financing trading strategy with zero initial cost can generate positive payoffs with positive probability whilst negative payoffs are impossible). I guess a weaker form would be the law of one price.

Are today's financial markets close to be in equilibrium, complete and free of arbitrage (at least most of the time)? Of course, there are some frictions and nothing is as perfect as in theory but all in all, what is the academic consensus?

Equilibria: in the macroeconomic sense of aggregate equilibrium where all markets clear, markets are most likely never in any equilibrium but rather in constant flux between different equilibria, because the market clearing macroeconomic equilibrium always depends on real and also in short run nominal factors which constantly change. Hence it does not make sense to say whether the markets are currently in equilibrium or not. However, textbooks use the concept of such equilibrium because it is very useful way of describing states toward which the markets gravitate - you can imagine them as a screenshot of a continuously playing movie.

Completeness: this condition, in Arrow-Debreu sense, requires there to be very small or non-existent transaction cost and perfect information, this is almost definitely not satisfied in many cases and consequently you will see a lot of research that looks at what happens when there are transaction costs and information asymmetries in markets.

Arbitrage: empirically arbitrage opportunities are definitely present in financial markets but they are usually small (i.e. no million dollar bucks left laying on a side walk) and they tend to disappear really quickly for example many studies of covered interest parity in foreign exchange markets show that arbitrage opportunities usually don’t last more than few hours, sometimes even seconds and arbitrage opportunities lasting few days are already considered long (Mark, 2000). History also shows longer arbitrage windows but they usually come from times where information technology was not as sophisticated as now. So you could say that the condition of no arbitrage in real life holds most of the time approximately.


If I find the GDP deflator (i.e., price level) for a given time series, can I then find real consumption given nominal consumption? I've found the gdp deflator for a country over a 25 year span. If I have nominal consumption for the same time series, then I just have to divide nominal consumption by the p-level I found with the deflator right?

GDP deflator deflates GDP. If you want to deflate consumption, you'd most likely want to use the consumer price index (CPI) to deflate. GDP deflator would take into account, for example, the price level faced by producers (PPI).


Difference between a Data Scientist and an Econometrician? I've noticed there are a lot of similarities between data scientists and econometricians and I'am having a hard time telling the difference between the scope of each field and what problems they attempt to answer.

What are the primary differences between a data scientist and Econometrician?

A central question of econometrics has been the question of causality: does X cause Y? Does the minimum wage cause employment? Do taxes cause capital flight?

Data science has been mostly concerned with the predictive power of its models: how can we sort pictures according to their contents? how can we sort customers according to their creditworthiness?

The two questions of causality and prediction are related: if we find a great predictor of customers' creditworthiness, does it mean that that characteristic causes creditworthiness? The answer is tricky, because excellent predictors are actually not always causal determinants of the outcome. If, for instance, high income students tend to cluster in good schools, this does not necessarily mean that household income causes student achievement. A perhaps more controversial example is that ethnicity predicts (i.e. is correlated with) a range of outcomes (education, health) but does not necessarily cause them.

Data science has become central to business applications (from Google Photos to face recognition in Facebook) but not so much to public policy analysis. In contrast, econometrics is ubiquitous in policy analysis.

There is some convergence though. A better, and more elaborate answer on how econometrics recently borrowed from data science is provided by Susan Athey (Sep 2016 podcast). http://www.econtalk.org/archives/2016/09/susan_athey_on.html


Do price controls increase competition? Question: Is there any way to show that price controls might increase competition?

I'm trying to find a way, any way, that price controls might lead to an increase in competition between companies. I've done a few hours of research, mostly just reading long PDFs, but I'm having difficulty finding any sort of analysis that shows that price controls increase competition among companies.

The best I've gotten is that some people argue that price controls are a kind of competition law (?) which are designed to increase competition. But that's not really a warrant for the question.

A classic example, related to the point made by @GuyLouzon, is access pricing in network industries.

A telephone network is a natural monopoly. The modern approach to regulating this market is to force the monopolist owner of the network infrastructure to allow competing telecomms services firms to buy access to the network. This allows those rivals to offer competing retail telephone services using the monopolist's infrastructure rather than having to build their own.

The price that the incumbent charges firms for network access is regulated. Ensuring that monopolist does not set the network access price too high means that downstream firms can profitably buy access and thereby enter the market for retail telecommunications services. While the fundamental natural monopoly problem has not gone away, regulating the access price at least helps to ensure the retail telehone service market is not also monopolised.

If you would like to read more about this example, I'd recommend "Competition in Telecommunications" by Laffont & Tirole.

One can easily build a model of this situation. Suppose that retail telephone firms earn profit $\pi_n$ when there are $n$ firms in the market. Make the standard assumption that $n\pi_n\leq m\pi_m$ whenever $n>m$ (industry profit is lower when there is more competition).

The wholesale firm sets a price, $p$, for access to its network. Firms will choose to buy access so long as $\pi_n>p$. The highest price the monopolist can charge if it wants to serve $n$ firms is therefore $\pi_n$. Thus, the monopolist can serve one firm and earn profit $\pi_1$, serve two firms and earn profit $2\pi_2$, serve 3 firms and earn $3\pi_3$, etc. Because $n\pi_n$ is decreasing in $n$, the monopolist optimally sets $p=\pi_1$ and grants access to a single firm, inducing a downstream monopoly.

A policy that constrains $p<\pi_2$ would, on the other hand, induce more than one firm to enter downstream and thus result in more downstream competition than when the price cap is not in place.


With whom does the US have the largest deficit using the trade in value-added (TiVA) methodology? The US has the largest gross trade deficit with China. But some economists have argued that this is misleading, e.g. using the iPhone as an example in which most parts don't orginate in China.

There's actually a more academic version of this analysis called "trade in value-added", which can be roughly summarized as:

  Global imbalances: accounting for trade in value-added (specifically accounting for trade in intermediate parts and components) and taking into account “trade in tasks” does not change the overall trade balance of a country with the rest of the world – it redistributes the surpluses and deficits across partner countries. When bilateral trade balances are measured in gross terms, the deficit with final goods producers (or the surplus of exporters of final products) is exaggerated because it incorporates the value of foreign inputs. The underlying imbalance is in fact with the countries that supplied inputs to the final producer. As pressure for rebalancing increases in the context of persistent deficits, there is a risk of protectionist responses that target countries at the end of global value chains on the basis of an inaccurate perception of the origin of trade imbalances.

This does look like quite an involved method to follow for all imports though. My question for here is: with whom would the US have the largest trade imbalance if one uses the "trade in value-added" (possibly recursively) to track US imports to their components?

The OECD provides value-added trade statistics from 2005-15. The table below, compiled by the Federal Reserve Bank of St. Louis, shows the U.S. trade balance in 2015 for several major trading partners in terms of gross trade and value-added trade.
Using the trade in value-added methodology, the U.S. trade deficit with China is reduced by 13 percent but still the largest.

Another figure that shows the U.S. trade balance from 2005-15 with several major trading partners in terms of real gross trade and real value-added trade.


How can markets can be highly efficient, even if they make errors? A Random Walk Down Wall Street (2015 11 ed, but an 2019 ed. is upcoming). p. 105 Middle.

    $\color{red}{\text{Markets can be highly efficient even if they make errors.}}$ Some are doozies, as when Internet stocks in the early 2000s appeared to discount not only the future but the hereafter. How could it be otherwise? Stock valuations depend upon estimations of the earning power of companies many years into the future. Such forecasts are invariably incorrect. Moreover, investment risk is never clearly perceived, so the appropriate rate at which the future should be discounted is never certain. Thus, market prices must always be wrong to some extent. But at any particular time, it is not obvious to anyone whether they are too high or too low. The evidence I will present next shows that professional investors are not able to adjust their portfolios so that they hold only “undervalued” stocks and avoid “overvalued” ones. The fact that the best and the brightest on Wall Street cannot consistently distinguish correct valuations from incorrect ones shows how hard it is to beat the market. There is no evidence that anyone can generate excess returns by making consistently correct bets against the collective wisdom of the market. Markets are not always or even usually correct. But NO ONE PERSON OR INSTITUTION CONSISTENTLY KNOWS MORE THAN THE MARKET. [I bolded]

Can someone please distinguish more clearly how $\color{red}{\text{'[m]arkets can be highly efficient even if they make errors'}}$? I don’t understand the differences between market efficiency and price correctness.

The theory of efficient markets is that no one can make rewards that are in excess of the level reflected by the risk they are taking, except by random chance.

So if markets can make mistakes, but not in a way that allows anyone to make disproportionate rewards except by random chance, then they can be highly efficient.


Financial Economics and Corporate Finance In pg 4 of Principles of Corporate finance (Allen et al, 2017), in the context of explaining financing decisions are less important that investment decisions, there is a line that says:

"Financial managers say that 'value comes mainly from the asset side of the balance sheet'"

What does this mean? Why asset side? Why not shareholders equity on balance sheet?
The text did use an example of Microsoft, where they point out that successful companies finance R&D by reinvesting existing cashflow.

This is referring to a general principle in finance. What matters most is the assets you have and the things you invest on, not really the way you finance these investments.  In the balance sheet, the asset side has the information about the assets you have, while the liability side has information about how you financed them.

This general principle comes from a very powerful idea called "the efficient markets hypothesis". Miller and Modigliani showed that if the financial market is perfect (basically if it has no private information, transaction costs, taxes, etc), the overall profitability of an investment does not depend on how it is financed!

Of course, markets are not perfect. However, many many studies have documented that most of the time, the financial market is pretty close to being perfect, especially in developed economies like the  US. So financing decisions do matter, but investment decisions matter a lot more.


What is consumer debt and how could it make businesses vulnerable? Consumer debt is the obligation to pay for the credit that the customers have used. But, how can tapping out more credits make business more vulnerable?

In the long run, consumers cannot spend more than they earn. In the short run they can spend more, increasing consumer debt, or they can spend less, decreasing consumer debt.

There is a limit to how much consumers are allowed to borrow. The limit is determined by consumer income and the interest rate.

If consumer debt increases faster than consumer income then eventually consumer debt will peak; bank and credit companies will stop lending more money causing a halt to the spending spree.

Note that consumer credit tends to increase market fluctuations. When times are good more people have jobs and consumers with jobs are allowed increased credit. When times are bad you see the opposite.


Comparative statics of a maximum Suppose that $f(x,a)=\max[g(x,a),h(x,a)]$ and $x^{∗}=\arg\max_{x}f(x,a)$.
Also, let $x_1(a)=\arg\max_{x}g(x,a)$ and $x_2(a)=\arg\max_{x}h(x,a)$.
If both $x'_1(a)$ and $x'_2(a)$ are nonnegative, under what conditions is $\frac{dx^∗}{da}\ge 0$?

As Jesper Hypel has pointed out in a comment, the derivative need not exist. Actually, the function $a\mapsto x^*(a)$ need not even be nondecreasing. Here is an example: Let $g$ and $h$ be given by $g(x,a)=-(a-x)^2$ and $h(x,a)=a-(a-1-x)^2$, respectively. Then $x_1(a)=a$ and $x_2(a)=a-1$. Moreover, $f\big(x^*(a),a\big)=g\big(x_1(a),a\big)$ for $a<0$ and $f\big(x^*(a),a\big)=h\big(x_2(a),a\big)$ for $a>0$. As a consequence, the function $a\mapsto x^*(a)$ jumps down after $0$.


Taxes levied on sellers = Taxes levied on buyers All page numbers refer to Principles of Microeconomics, 7 Ed, 2014, by NG Mankiw.

  [p 125:] Taxes levied on sellers and taxes levied on buyers are equivalent.
  [p 156:] ... When a tax is levied on buyers, the demand curve shifts downward
  by the size of the tax; when it is levied on sellers, the supply curve shifts upward
  by that amount. In either case, when the tax is enacted, the price paid by buyers
  rises, and the price received by sellers falls. ...

What's the intuition?  I already proved this 'surprising conclusion' with a supply-demand graph (p 130, Question 6.7) and so ask NOT about a graphical or rigorous math proof.
I understand the para below, but don't comprehend how it intuitively explains the supply-demand mechanisms (the quote below concerns only how the tax is sent to the government)?

  [p 125:] The equivalence of these two taxes is easy to understand if we imagine that the
  government collects the $0.50 ice-cream tax in a bowl on the counter of each ice cream
  store.' .....

Whether the tax is imposed on the buyer or the seller doesn't make any practical difference. It's just an accounting detail. One might even say, it's just an accounting trick.

I haven't read this particular book but I imagine what he was driving at with the "bowl" example was this:

Suppose that instead of filing tax forms, etc, the government collected taxes on the spot: There is a tax collector standing beside the counter with a bowl to collect the tax money. The price not including the tax is \$10. The government charges a \$1 tax.

So, scenario 1: The government declares that the buyer must pay the tax. So when you buy the product, you hand the seller \$10, and then you drop \$1 into the tax man's bowl.

Scenario 2: The government declares that the seller must pay the tax. So the seller increases the price to \$11. When you buy the product, you hand the seller \$11. He then drops \$1 into the tax man's bowl, and keeps the remaining \$10.

Either way, the buyer has \$11 less cash, the seller has \$10 more cash, and the tax man has \$1 more cash. Whether the buyer hands the tax money directly to the tax man, or the buyer hands the money to the seller and the seller hands it to the tax man, makes very little practical difference.

We could discuss what portion of the tax is really born by the seller and what portion by the buyer. If today the tax is zero and tomorrow the government begins imposing a \$1 tax, it doesn't necessarily follow that the seller can continue to collect \$10 and the buyer will pay \$11. Increasing the cost to the buyer to \$11 would mean that fewer units are sold -- some number of people will not buy or will not buy as many for \$11. So the seller may conclude that to maintain sales sufficient to maximize his profits, he must drop the price to, say, \$9.60, so with tax the total price is now \$10.60, and the seller is paying 40 cents of the tax while the buyer is paying 60 cents. Or he may have to drop the price further, or not drop it as much. But whatever the optimum price ends up being, whether the government nominally collects the \$1 from the buyer or from the seller has nothing to do with how the tax is really being split between buyer and seller.

There is the technicality that there may be costs involved in record-keeping or other mechanics of the tax. Like if the seller is required to collect the tax and turn if over to the government, then the seller has additional book-keeping costs. If the buyer is expected to keep records and pay the tax, then the buyer has book-keeping costs. But this isn't the main point.


From the aggregate supply equation to the definition of inflation rate By definition the  inflation rate is $$\pi=\dfrac{P-P_{-1}}{P_{-1}}\cdot100\%$$ or could be defined in terms of the consumer price index CPI, but in this case I think the former is the one to consider.
From the book Macroeconomia by Blanchard, Amighini, and Giavazzi,
the aggregate supply equation is \begin{equation}
P=P^e(1+\mu)F(u^-,z^+)
\end{equation}
From here, and taking a specific $F:$
\begin{align}
    P&=P^e(1+\mu)e^{-\alpha u+z} \nonumber\\
    \ln(P)&=\ln(P^e)+\ln(1+\mu)-\alpha u+z  \nonumber\\
    \ln(P)-\ln(P_{-1})&=\ln(P^e)-\ln(P_{-1})+\ln(1+\mu)-\alpha u+z \nonumber\\
     \ln(P)-\ln(P_{-1})&=\ln(P^e)-\ln(P_{-1})+\mu-\alpha u+z\dots \text{due to $\ln(1+\mu)\approx\mu$ because $\mu$ is close to $0$ } \nonumber
  \end{align}
Thus \begin{align}\pi=\pi^e+(\mu+z)-\alpha u \end{align}
Why? For example how to pass from  $\ln(P)-\ln(P_{-1})$ to $\pi$?
We know that $\ln(P)-\ln(P_{-1})=\ln\dfrac{P}{P_{-1}}\fbox {=?}\dfrac{P}{P_{-1}}-1$
Thank you in advance

This is because for small values $x$, $$\ln x_{t+1} - \ln x_{t} \approx \frac{x_{t+1}-x_{t}}{x_{t}}.$$
This holds since the growth rate $g$ can be expressed as follows:
$$g= \frac{x_{t+1}-x_{t}}{x_{t}} \implies x_{t+1} = (1+g)x_{t}$$
taking logs we get that:
$$ \ln x_{t+1} = \ln (1+g)+ \ln x_{t} \implies \\ \ln x_{t+1} -\ln x_{t} = \ln (1+g) $$
Finally, for small values of $g$ we get that $\ln (1+g) \approx g$ thus we get:
$$\ln x_{t+1} -\ln x_{t} \approx g $$
By the same token for small values of inflation (and inflation typically will take values $\pi<0.1$) it is completely reasonable to just define inflation as $\pi = \ln P_t - \ln P_{t-1}$ (although it would be more appropriate to use $\approx$).


What mathematical structure or formalism can be used for modelling strategic investment decisions of agents in the context of competition? I'm trying to analyze the decisions of agents in investing scarce resource like time and money into developing their product/service/offering in the presence of competition. I need to know what kind of mathematics can be used to describe this process, either analytically (preferably) or in the form of an simulation algorithm. Also, I'm working in the context of the Web and the Internet.

For example:

Suppose each agent has two possible strategies:

Strategy A, or S(A), consists of a high investment cost, which has high returns in the absence of competition or other possible adverse factors.
Strategy B, or S(B), is a low investment strategy, which comes with smaller returns that are more predictable.

In each possible strategy, there is an option of which products to produce, and also how much to specialize by differentiating these products. Each product has its own cost/benefit profile, i.e. the level of demand for each differentiated version of the product is different, and so is the cost of producing it.

I would like to know how to model situations like the above.

Specific questions I have in mind are:

Is there an equilibrium that can be reached?
What are the conditions under which each agent would have the incentive to invest as much as possible?
What does the equilibrium look like, in terms of levels of investment and specialization.

Example reference:

"Media, aggregators, and the link economy: Strategic hyperlink formation in content networks" by Chrysanthos Dellarocas, Zsolt Katona, William Rand

I found the above paper very illuminating. I would like to find more work along these lines, but haven't been very successful at tracking down any.

Disclaimer: I am NOT presuming to have an answer to this question but given the lack of any sort of input I am throwing some ideas out there so hold your horses with your downvotes.

It sounds to me like this is something you can model as a game of incomplete information. As a very very basic set up (see for example Moulin for a detail on games), in a Bayesian form a game specifies:

The set of $N$ playes
The set of pure strategies $X_{i}$ for each player $i$
The set of types $T_{i}$ for each player $i$
The set of beliefs of each player $i$, represented by a probability distribution $\pi_{i}(.|t_{i})$ over $T_{N \text\i}$: one distribution for each possible type of player $i$
The payoff function $u_{i}(x;t)$ for each player $i$, where $x \in X_{N}$, and $t \in T_{N}$

A Bayesian equilibrium is described by a mixed strategy (combination of strategies) for each player, conditional on his type: $s_{i}(t_{i}) \in \Delta(X_{i})$. The equilibrium property then is $$ \forall i,t_{i} \in T_{i}, \forall s^{'}_{i} \in \Delta(X_{i}): $$
$$ \Sigma_{t_{-i}\in T_{N \text\i}} \pi_{i}(t_{-i}|t_{i})u_{i}(s(t);t) \geq \Sigma_{t_{-i}\in T_{N \text\i}} \pi_{i}(t_{-i}|t_{i})u_{i}((s^{'}_{i},s_{-i}(t_{-i}));t) $$

were the following notation is used $$ s_{j}(t)=s_{j}(t_{j}); s(t) \in \Pi_{j \in N \text\i} \Delta(X_{j})$$

Now, if the sets $X_{i}$ and $T_{i}$ are finite, the game possesses at least one Bayesian equilibrium. (This is a direct consequence of Nash's theorem and you can easily find a prove online so I will not get into this).

Now, this is all very abstract, sort to speak. But applied in a simple way to the example you provided your components are quite straightforward: you have some set of agents who will make an investment decision with some goal in mind (obviously here is to maximize some payoff function), and a stochastic component in the returns that each investment offers. Types may refer to some heterogenity that you might want to impose on your players, and beliefs may refer to their knowledge of states of the world or other player's decisons. Now, you ask whether an equilibrium can be reached. As stated before (and of course this is just to provide you with a basic set up) under certain assumptions the existence of an equilibrium is shown. Now, figuring out that equilibrium, as well as the as the conditions under which each agent would have the incentive to invest as much as possible depends entirely on the complexity of the strategic interaction you model, and on the assumptions you make (among admittedly many others). You can make it as easy (e.g. take to agents with some payoff functions, impose known priors, make it a one shot game, and so on and you will easily find some equilibria, as uninteresting as this may be), or as difficult as you'd like it to be. In general, game theory is concerned with agents that interact, so it seems to me that there is a lot of use for you in this area given what you want to model. As you can see the notion of utility maximization is there, so this is not an issue.

Now, as far as the paper you mentioned and the general area I am not able to provide any help. But there are many papers out there on investment decisions and competition from that use game theoretical settings (see for example the work of Smit and Ankum, and i'm sure some key search such as game theory investment competition will bring you many more).

I am aware this is not a satisfying nor complete answer, but I am sure that if you look around at literature even outside your topic you will get a good idea on how you can develop a model (in my own experience this requires a lot of creativity and you often require "inspiration" from the work of others even if it does not necessarily relate to your topic so I would recommend you look at papers in general to gain some insight into the modelling component)

This other papers may provide an interesting perspective on competition (and investment) and game theory models:

https://link.springer.com/chapter/10.1007/978-3-319-05951-8_37

https://www.researchgate.net/profile/Lee_Cooper/publication/227442687_Modeling_Asymmetric_Competition/links/09e415140c23842b7f000000.pdf

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1745-6622.2001.tb00332.x

https://ieeexplore.ieee.org/abstract/document/977512

https://ieeexplore.ieee.org/abstract/document/4438796

I hope this is of some help


Why is GDP growth so important that central banks are trying to impose negative interest rate? BoJ just announced its negative interest rate policy in an effort to drag Japanese out of a deflationary mindset and hoping to shift the demand curve to boost the economy. However, my question is, if people don't want to spend, why bother?

Consider the following simple model of an ideal economy. Suppose this economy has a maxium output is 100 units of bread. Further suppose it is a one-factor economy, i.e. all output is produced by labor. Suppose everyone in the country is considered a labor and there is a total of 100 units of labor in the economy. Suppose the most comfortable level of consumption of bread by people is a total of 10 units. Then my question is why would the cnetral banks impose policies to force people to demand more of it when they do not want to? It only made sense under the following circumstance: since only 10 units of bread is needed, maybe only 10 units of labor is employed to do all the production and the rest of the people is not employed, which means they do not have any money to buy the bread. In this case, it made sense for government to raise demand so the rest of the people can have a job and earn money, and therefore spend money, which further helps the economy. However, in Japan's case, unemployment rate looks fine, around $3.5\%$ as I checked yesterday, partly because Japanese companies are reluctant in cutting employees. Connecting to the simple model I had beforehand, Japan's case is like having all the 100 units of labor employed to produce 10 units of bread, althought they really only needed 10 units of labor. This implies that all the people are getting what they want.

So why is there anything to worry about that the central bank of Japan even start to adopt these agressive monetary policies?

Central banks do not try to increase growth. Instead, they commonly have two targets: 1) to maintain price stability and 2) to minimize the output gap. Let us ignore the first goal for a moment. The minimization of the output gap implies that if output falls short of what the central bank believes to be the natural level of output, then it will try to use economic policies which promote growth.

In your example, 10 units of bread being produced seems to be the natural level of output. If the central bank sees that the economy produces only 9 units of bread, it will try to stimulate the economy with low interest rates. If it observes 11 units being produced, it will raise interest rates to reduce the output. (Note that this is still possible even if firms produce 9 units of bread using 10 (or even 100) units of labor, i.e. if firms are not cutting employees efficiently.). Thus, if the central bank of Japan currently has low interest rates, it believes that current output falls short of the natural level of output, despite low unemployment rates.

You may still ask why the central bank tries to minimize the output gap in the first place. Why should it not just focus on price stability or why not even do something random. It turns out that up to second order approximations, an interest rate policy which minimizes a combination of price volatility and output gap volatility maximizes people's welfare in so-called new-Keynesian models. This was first shown by Woodford and Rotemberg in several articles:

  Woodford, Michael. "Inflation stabilization and welfare." B.E. Journal of Macroeconomics, 2, (2002).

  Woodford, M. (1999). Optimal Monetary Policy Inertia. Retrieved from http://www.nber.org/papers/w7261

  Rotemberg, J., & Woodford, M. (1999). Interest rate rules in an estimated sticky price model. Monetary Policy Rules. Retrieved from http://www.nber.org/chapters/c7414.pdf


Does the PBoC work like the Fed in having meetings where they discuss changes in monetary policy? The Fed has a specific schedule to its meetings about changing the Fed Funds Target Rate, does the PBoC perform the same as well?

I haven't posted here in a while, but I saw this question and thought I ought to respond.

No. That is the short answer to your question.

After having been living and working in China for a year, I realized that nothing about its economy can be viewed separately from politics and social psychology of mainland Chinese population. Simply put, everything is different.

People's Bank of China is defacto a part of executive branch of government. It sets monetary policy decisions in accordance with the government's social and economic policies, which are ultimately curated by the Chairman of the Communist Party (a.k.a the President). This makes it very predictable. If the growth target is set to 6.5%, the PBOC will provide enough liquidity to hit the target, pretty much at all costs.

In China's political culture collective debates are ubiquitous, but when something is being discussed, everyone tends to converge on the leader's opinion very quickly.

However, one can only speculate how PBoC board makes their decisions. It is not very transparent, like most of the political culture in the country. Everyone seems to be happy though that it is, at least, predictable.


Modern Portfolio Theory Vs Marginal Utility Theory I'm currently trying to wrap my head around modern portfolio theory and would love a simple explanation on how it differs from a marginal utility model (if at all).

As I am understanding it, MPT allows you to create a portfolio of assets in optimal quantities at a given risk factor.  However, isn't this basically the same as judging the marginal risk return of each asset and then funding each to its relative "risk return" (e.g. fund Asset A until marginal return is less than asset B, then fund asset B, fund asset B until marginal return is less than C, then fund C).

Welcome to EC.SE! Hopefully this will help.

I don't think this characterization is correct. These are not opposing theories. Modern portfolio theory is based on the idea that people choose portfolios to maximize their utility. After that, it depends on the specification of utility. As a simple example, a person with a quadratic utility function will choose a portfolio that is mean-variance efficient. The particular portfolio allocation will depend on the risk-aversion parameter in the utility function.

For examples discussed in other questions on the site, see Calculating the optimal portfolio for an investor with quadratic utility and
Portfolio choice problem of a CARA investor with n risky assets.


When can we expect China to become a developed country? China has the 2nd largest GDP in the world and also has a high GDP growth rate. According to Richard Lynn's IQ and Global Inequality, the Chinese have a mean IQ of 105, which is one of the highest in the world. So, it just seems to be a matter of time before China becomes a developed country. The only question is when: Will it be 10 years from now? 20 years? 50 years?

Just because a country has a lot of resources (including human capital), it does not necessarily form a path for a developed nation. China is also very involved in military conflicts. It just takes one war for a country to drift back decades. Also, you can't ignore the brain drain problem.

With that, "developed" nation is a fluid definition. What does that mean? Is the US developed? Thousands of US citizens don't have electricity yet. No clean water. Massive health care and student loan crisis. Not to forget the trillions of dollars of debt. But many would consider the US a developed nation. If you are basing "developed" as in militarily-wise, then China will soon be a developed nation, if not already.

However, if you account for freedom of speech and all, then China is still decades behind from its neighbors like India.

Hope that helps!


What are the economic after effects of a 10% pay cut to all state employees? I am attempting to ask this question as impartially as possible, setting aside the political and social ramifications.
Given: the shut down that began on March 2020, as a response to a contagious virus, known as COVID-19, decimated state and local tax revenues.
The Governor in reaction to discovering his tax revenues were well below his expectations asked all state employees to take a 10% pay cut.  Additionally, he is asking for teachers to be retrained so that they can video conference their classrooms.
Meanwhile, some local politicians are reacting by drafting a stimulus bill.
I understand businesses respond to a crisis like this by not hiring as many employees and reducing salaries until they can ramp back up to pre-crisis mode.
My question is which of the following approaches is better for the economy at large, and what are their economic effects and why? And how long would it take for each one of these approaches to stabilize the economy?  Is there another approach that I am missing?

A) 10% across the board cut in salaries to state employees
B) A laissez-faire response
C) A stimulus bill

From a simple and purely theoretical point of view:
Option (A) would give you a lower level of consumption and income state-wide and generally is a budget-balancing method, not an economic equilibrium-producing method. Option A is surely on the opposite side of the "policy target" spectrum, compared to B and C.
Option (B) is considered best when prices are not "sticky" - that is they drop quickly on a recession. Current data shows ( https://fred.stlouisfed.org/series/CP0000USM086NEST , https://ycharts.com/indicators/us_monthly_gdp ) that prices in the US have dropped but at a very lower speed than GDP, so I'm inclined to call them sticky.
(To put it bluntly: When that happens, people lower their food and medicine consumption, they get sick and die. Not the best idea. )
When that doesn't happen, and prices adjust quickly for the amount of income lost, equilibrium ensues. I'm inclined to say that this has never happened in the history of the human race, apart from closed experiments. At the very least, I know of no such case in modern economic history (that is, after the Renaissance till now).
Option (C) is the approach that is thought of as most successful when sticky prices occur, because a stimulus bill will put money in people's hands and consumption and production will return to their "equilibrium" level faster.
Critics of option (C) -which seems to be the most fitting at the time- cite concerns over inflation rates higher than those prior to the crisis, and also, there are problems concerning the distribution of money, that is, if no one pulls strings over the system. We wouldn't want to be ignorant over political matters.


GDP versus GNP. Which is a better gauge of a country's economy size and prosperity? Country A's GDP is larger than Country B's GDP. However, Country B's GNP is larger than Country A's GNP. Which country has a stronger economy? Which country's citizens is more likely to enjoy prosperity?

I think the safest answer is “it depends.”

Firstly, the terms “stronger economy” or “enjoy prosperity” are subjective, so I think if you want a stronger answer, you would need to tighten that part of the question up, with more concrete phrasing.

Secondly, we would need to pin down the reasons why GDP and GNP diverge.

Ireland is a well-known example, where GDP is much larger than GNP. This is usually ascribed to multinational corporations funnelling revenue through Ireland to take advantage of low corporate tax rates. In this case, GNP is viewed as being more representative of the state of the Irish economy, since those corporate revenues have very little to do with the Irish economy.

I am unaware of examples where GNP is much larger than GDP, but I believe that it would be easy to create theoretical examples where GNP is telling us very little about domestic economic conditions. (A country with large ownership of foreign assets, where production and income is almost entirely the result of the economic conditions facing those foreign subsidiaries.)

In other words, you need to know exactly what question you want to answer, and see whether GDP or GNP is closer to what answer you need.


What is the optimal forecast for the following moving average model? Suppose that we have an MA(1) model where $y_t=\epsilon_t-\frac 1 2 \epsilon_{t-1}$ and $\epsilon_t$ are i.i.d.
I want to find an optimal forecast of $y_{T+1}$ in the form of $Cy_T$ where C is a constant. What is the best choice of C?
I have computed that

$\begin{aligned}
&y_{T+1}=E(y_{T+1}|I_T)=E(\epsilon_{T+1}-\frac 1 2 \epsilon_T|I_T)\\
&=E(\epsilon_{T+1}|I_T)-\frac 1 2E(\epsilon_T|I_T)\\
&=0-\frac 1 2\epsilon_T=-\frac 1 2 \epsilon_T\\
&\epsilon_T=y_T+\frac 1 2y_{T-1}+\frac 1 4 y_{T-2}+......+{1\over 2^{t-1}}y_1+0\\
&y_{T+1}=-\frac 12 \epsilon_T=-\frac 12y_T-\frac 1 4y_{T-1}-\frac 1 8 y_{T-2}-......-{1\over 2^{t}}y_1
\end{aligned}$

Your equation $y_{T+1} = E(y_{T+1}|I_T)$ clearly does not make sense.

In any case,
$$
E(y_{T+1}|I_T)
$$ is the best one step ahead forecast conditional on $Y_1, \cdots, Y_T$, whereas you're asked to find the best linear one step ahead forecast given $Y_T$. They are in general not the same.
The best linear forecast for $Y_{T+1}$ given $Y_T$ is
$$
\mu + \frac{\gamma(1)}{\gamma(0)} (Y_T - \mu),
$$
where $\mu$ is the unconditional mean of the series and $\gamma$ is the autocovariance function. In your case, $\mu = 0$ and $\gamma$ is that of a MA(1) series ($\frac{\gamma(1)}{\gamma(0)} = \frac{\theta}{1+\theta^2}, \; \theta = \frac12$).

In general, the best linear forecast for covariance-stationary time series can be obtained by solving the Yule-Walker equations.

For best linear forecast, the assumption that innovations are i.i.d. is extraneous.


What are the drivers of the price of land? I'm interested in what factors lead to changes in the price of land. I'm interested in this question in general across a whole country, say (as opposed to land in particular places).

So I'm interested in this question, because I'm interested to build a stock flow consistent model of 'Georgist' effects: integrate georgist understanding into a  post keynesian model framwework.

As a follow up issue, I'm also interested to what extent the macro drivers of land value are shared with shares in companies that have extensive market power (e.g. coca cola). Can land be thought of as an archetype of other 'monopoly' assets.

Because of population growth, savings, and technological innovation the stock of capital and augmented labor are growing over time. These forces raise the marginal product of land. Unlike capital, the supply of land is fixed. This means that the supply of land can't increase in response to the rising marginal product. Investors have to be indifferent between investing a dollar in holding land and a dollar of productive capital. Capital is paid its marginal product ($r$). For investors to be indifferent, each dollar buys $1/P$ worth of capital, and that fraction of a unit of land returns $r$. One unit of real estate actually returns $rP$. So, over time, the forces that raise the marginal product of labor and capital, population growth, capital deepening, and technological innovation, raise the marginal product of land and therefore push up the price because the quantity cannot adjust.

This assumes we can abstract from issues of taxation and depreciation. We can handle those details but they don't change the basic story. What is more important issue is that land is in fixed supply and capital is not.

There are some innovations that increase the literal or effective quantity of land: e.g. landfill, multi-story construction, enhancements in in agricultural productivity. More accurately, in practice, the quantity of land can adjust can adjust in a limited way, more slowly, and at greater cost than physical capital. This preserves insight above: quantities do not adjust to equate the marginal product of physical capital and land, and so the price must do so instead.


Jordi Gali optimal price setting Monetary Policy, Inflation,
and the Business Cycle,
Jordi Gali,
Page 45
http://elib.peaceland.edu.ng:8383/greenstone3/sites/localsite/collect/peacelan/index/assoc/HASH0139.dir/doc.pdf

How does "ψ " convert to "MC"?

You can say Drago Bergholt solutions for Gali
http://bergholt.weebly.com/uploads/1/1/8/4/11843961/the_basic_new_keynesian_model_-_drago_bergholt.pdf
but Drago also didn't explain this change. (page 13 )

Sincerely

The third line of p.45 denotes:

$MC_{t+k|t}=\psi_{t+k|t}/P_{t+k}$

but this is probably not the answer you are looking for.

Going from eq. (9) to (10), multipliy the right-hand side by $\frac{P_{t+k}}{P_{t+k}}=1$ (you can always multiply by one), the numerator goes into $\Pi_{t-1,t+k}$ (together with $P_{t-1}$), the denominator goes into $MC_{t+k|t}=\psi_{t+k|t}/P_{t+k}$.


Some basic questions on infromation asymmetry How does asymmetric information and bid-ask spread are connected? For instance, economic theory tells us that, the more the problem of information asymmetry, the higher the spread is. What is the intuition behind it? Can this change?
Furthermore, what is the difference between asymmetric information, incomplete infortation and imperfect information?

You want to look at a previous post: Imperfect vs incomplete information .

Basically complete information relates to the assumption that players have common knowledge about the game and the utility function of all players. However, they may not know the cards that a player was dealt with, or there might be an action taken by some player that they do not observe.

Perfect information relates to observing all moves and strategies available to a player, but perhaps not knowing the utility or being uncertain about the game. Examples of perfect but incomplete information are a bit harder to imagine, but the examples I like to think of are board games where players have secret goals or missions.

Asymmetric information intersects with both of these classifications. A game has asymmetric information if one player has more information than another player. That is, perhaps one player has complete information and the other one has incomplete information, or perhaps one player has perfect information and the other one has imperfect information. There are many examples here, perhaps one of the most famous ones is the market for used cars, where the seller usually knows better the condition of the car than the buyer. I.e. the information is asymmetric.

Your question about the bid-ask spread is a bit too general. I think that people have used economic theory to rationalize the existence and size of the bid-ask spread. To my knowledge, there are many reasons why this spread exists, and some of these mechanisms have nothing to do with the information.

For example, consider the market for currencies. If in a small town there is a single bank that exchanges currency X for currency Y and vice versa. The bank has market power and can get away with a large bid-ask spread. Even if there is complete, perfect and symmetric information, people might prefer to exchange currencies at a relatively large ask and a relatively low bid, instead of traveling to the next town to find a  different bank.

Pertaining information, I can think of traders in the financial market exchanging currencies. If there is a dealer with better information as to where to find more favorable exchange rates, it will have more leeway to increase their own bid-ask spread. Since the their clients might not know there are better options out there in the market.


What is the service equivalent of an economic 'bad'? In general conversation, the terms "goods and services" are often used together. In economics, these terms have the following meanings:
Goods:

In economics, goods are items that satisfy human wants and provide utility, for example, to a consumer making a purchase of a satisfying product.

Services:

A common distinction is made between goods which are transferable, and services, which are not transferable.

In economics, a 'bad', is:

the opposite of an economic good. A 'bad' is anything with a negative value to the consumer

Is there any more specific term for a 'bad' (something with negative value to the consumer) which is strictly a service?

Any labor may be considered as a 'bad' (service) for which you are compensated by the firm by paying you salary/wages.
In modeling labor market, this is generally described in the opposite sense, i.e., consuming leisure. When you work you let go of leisure, which is a good. So that, technically, makes supplying labor (which is a service) as an economic bad.


How to prove convexity + quasilinear preferences imply concave utility? Let $\succsim$ be a strictly convex and quasilinear preference relation. It's defined over, say, $\mathbb{R}^2_{+}$ and is quasilinear on good 1.

So, $U(x_{1},x_{2}) = x_{1} + f(x_{2})$. How to prove that $f$ is a strictly concave function?

I'm solving problem 15.B.8 from MWG and I can't even understand what the solutions manual did!

Thanks a lot in advance for any hints and ideas!

Consider any $x_2'$ and $x_2''$ in $\mathbb{R}_+$. Without loss of generality, let $x_2'' > x_2'$. We can choose $x_1'=f(x_2'')-f(x_2') > 0$ so that $U(0,x_2'')=U(x_1',x_2')$. Let $\lambda(x_1',x_2')+(1-\lambda)(0,x_2'')$ be a convex combination of $(x_1',x_2')$, and $(0,x_2'')$. Since $\succsim$ is strictly convex and $U(0,x_2'')=U(x_1',x_2')$,
        \begin{eqnarray*}
 && U(\lambda(x_1',x_2')+(1-\lambda)(0,x_2''))  >  U(x_1',x_2') \\
 &\Rightarrow & U(\lambda x_1'+(1-\lambda)0,\lambda x_2'+(1-\lambda)x_2'')  >  \lambda U(x_1',x_2')+(1-\lambda)U(0,x_2'')\ldots(\because U(0,x_2'')=U(x_1',x_2')) \\
 &\Rightarrow &  \lambda x_1'+(1-\lambda)0 + f(\lambda x_2'+(1-\lambda)x_2'') >  \lambda (x_1'+f(x_2'))+(1-\lambda)(0 + f(x_2''))\\
  &\Rightarrow &  \lambda x_1'+(1-\lambda)0 + f(\lambda x_2'+(1-\lambda)x_2'') >  \lambda x_1'+(1-\lambda)0 + \lambda f(x_2')+(1-\lambda)f(x_2'') \\
    &\Rightarrow &  f(\lambda x_2'+(1-\lambda)x_2'') >   \lambda f(x_2')+(1-\lambda)f(x_2'')\end{eqnarray*}
Therefore, $f(\cdot)$ is strict concave.


What behaviour do different values of Relative Risk Aversion imply? What behaviour does it imply that Relative Risk Aversion of consumption, say R(c) is less than 1, equal to 1 or greater than 1?

I am reading an article of Diamond & Dybvig (1983), about Bank Runs, Deposit Insurance and liquidity and they assume a relative risk aversion greater than 1. Unfortunately, some of my microeconomics are a couple of years ago. I know the formulas and so on, but am unsure of the intuitive implications on behaviour for different values of Relative Risk Aversion.

In the D&D model, We deal with a 2 periods economy where the agent can invest in a short term project that yields $0$ or a long term project that yields $R$;

It is possible to prove that an optimal insurance scheme (such as the use of a mutual fund or a bank) flattens the yield curve.

It is possible to show that the optimal portfolio problem solves:

$max_{\{c_1\}}\{\lambda u (c_1) + (1-\lambda) u(\frac{c_2 R}{1-\lambda})$

where $\lambda$ is the share of impatient consumers who consume in period 1 (those who "face a liquidity shock"), and $R$ is the yield on long term projects. The solution to the problem above gives: $\frac{u'(c_1)}{u'(c_2)} = R$.

Under the assumption of $u'>0, u''<0$, the condition $-\frac{cu''(c)}{u'(c)} > 1$ is technical; it ensures that $\forall c$, the first derivative of $ cu'(c)$ is negative, or that $cu'(c)$ is decreasing in c. Given $R>1$ this implies that $Ru'(R)>1u'(1)$ and so given that $c_1 = 1, c_2=R$ we have that $\frac{u'(c_1)}{u'(c_2)} = \frac{u'(1)}{u'(R)} > R$ .

Going back to the FOC of the optimal porfolio problem, in order to reach the equality we therefore need to have $c_1^*>1$ and $c_2^*<R$ so that to insure that $\frac{u'(c_1)}{u'(c_2)}$ decreases enough. This means that an insurance scheme determines a flattening of the yield curve

The bottom line is that there is not much intuition on why the condition holds but you can easily think of why it has to hold (given standard assumptions on the utility function) by looking at $lim_{c\to x} (-\frac{cu''(c)}{u'(c)})$ when $x= 0$ and $x=\infty$.

For more details on the DD model look at: Tirole Jean, "The Theory of Corporate Finance"


When first differences contradict a regular regression regarding Investment vs Output relationship Im doing some research with  some panel data I have on firm output and investment.

I ran two equations.

$$y=\beta_0+\beta_1x+\beta_2x^2+\mu$$
$$\Delta y=\alpha_0+\Delta\alpha_1x+\epsilon$$

in R these equations were prodouced using these commands

   1) lm(output~investment+I(investment^2))
   2) lm(diff(output)~diff(investment))

The first equation had statistical significance on the 1% level, however with the first differenced data statistical significance was lost, and I ended up with rather high p-values as if there is no relationship between the two variables.

What is the interpretation of such results (i.e is there a relationship between investment and output in this data set) and which regression should I use in my research?

It's hard to say. Presumably you think you have some fixed (unobservable) omitted variables, which is why you ran fixed effects (FE) / first differences. So I see one of two things going on here:

The fixed omitted variables caused significance in the first regression, which means you do not have a significant effect. Do you have fixed omitted variables? If not, go with the first regression instead.
A quadratic model fits the data much better and hence you have significance in the first model, but not the second. In that case, you probably do have a significant effect. Is that true though? Why did you choose a quadratic specification? Does theory or the scatter plot suggest that is a good idea? If not go with the second regression.

My suggestion would be to try fixed effects/first differences  with the qudratic terms if you have good reason for both. Otherwise use the specification you think has the most merit.

Unfortunately, I am not familiar with R and cannot comment on the code.


Value of Statistical Life and Risk I have been reading a paper by Bove & Elia (2011), where they quote a definition of the value of statistical life from Bellavance, Dionne & Lebeau (2009). I have tried making my peace with the necessity of this concept, but I have been rather confused by the used definition. It says that the value of a statistical life is a function of the willingness to pay to reduce the risk of death and the risk of death itself. However, the risk of death reduces the value of a statistical life. Why would it do that? Is it a weight as in an NPV calculation, where the risk of default reduces the expected cash flow?

If you're willing to pay \$10 to reduce your risk of death by 1 in 1M, then your VSL is \$10M.†

Example. Say there are two bike helmets. One costs \$20 while the other costs \$30. The two helmets are exactly identical, except that the more expensive one has been proven to reduce your risk of death by an additional 1 in 1M.

If you're willing to buy the more expensive helmet, then your VSL is (at least) \$10M. (If you're exactly indifferent about buying the more expensive helmet, then your VSL is exactly \$10M.)

I'm not sure what you mean by "the risk of death reduces the value of a statistical life."

†At least under the simplest approach, which ignores other important considerations like risk aversion and the discrepancy between willingness-to-pay and willingness-to-accept.


Quasiconcavification Let $f_1, f_2$ be two smooth strictly-quasiconcave functions.
  Do there always exist monotone transformations $g_1,g_2$ such that the sum
  $g_1\circ f_1 + g_2 \circ  f_2$ is a strictly-quasiconcave function?

While this question is mathematic, its motivation is economic. Given two people with strictly-convex preferences, we would like to represent their common preferences by a social welfare function. One natural way to do this is to represent each person's preferences by some strictly-quasiconcave function and take their sum. However, the sum of strictly-quasiconcave functions is not necessarily quasiconcave. So it is interesting whether we can always find specific representative functions such that the sum is quasiconcave?

I already know that, without the "strictly", the answer is no. Here is an example:

Both functions are weakly-quasiconcave. When one function is increasing, the other is flat, and when the other is decreasing, the first is flat. Therefore, regardless of what transformations we apply to them, the sum will look like a wave and will  not be quasaiconcave.

Note: I asked this question on MathOverflow some time ago. Taneli Huuskonen came up with the negative example for weakly-quasiconcave function, but the question for strictly-quasiconcave function is still open. Due to the economic importance of quasiconcave functions, I thought someone here might have an answer.

In one dimension if the functions $f_i$ are bounded, e.g. defined on a compact interval, then yes. Take $g_1$ such that it makes $f_1$ almost constant, e.g. $g_1(x)=\frac{x}{N(\max f_1-\min f_1)}$ where $N>0$ is large. Take $g_2(x)=x$ Then $g_1\circ f_1+g_2\circ f_2\approx f_2$, which is strictly quasiconcave by assumption.

In one dimension a strictly quasiconcave function is either strictly increasing, strictly decreasing or single-peaked.


etymology of TFP (total factor productivity) In economics, total-factor productivity (TFP), also called
multi-factor productivity, is usually measured as the ratio of
aggregate output (e.g., GDP) to aggregate inputs.1 Under some
simplifications about the production technology, growth in TFP becomes
the portion of growth in output not explained by growth in
traditionally measured inputs of labour and capital used in
production.

https://en.wikipedia.org/wiki/Total_factor_productivity
How can factors "not explained by growth in traditionally measured inputs of labour and capital" come to be described as "total"? Etymology, please. It would seem like "not" implies "complements" and "complementary" so CFP sounds better than TFP.

It is total because it takes into account all output per all inputs - hence it is the total productivity of all factors.
Total factor productivity as that excerpt from Wikipedia states is given as:
$$TFP=\frac{\text{GDP}}{\text{All Factors}}$$
that is where the total comes from. In practice, the factors used in denominator are capital and labor, but they are measured very broadly to include other factors (e.g. human capital will often be 'hidden' in the labor variable). However, this being said some factors are very hard to measure (e.g. some economists consider entrepreneurship a factor of production), this is why some economists prefer to call it multifactor productivity (i.e. it is productivity of multiple factors but not all), as mentioned in the OECD glossary. However, it has nothing to do with complementarity which in economics has completely different meaning than what you imply in the question so CFP would make no sense.
Also, the passage you cite as reason for your confusion is actually not about total factor productivity but about growth in total factor productivity, i.e. not TFP but $\Delta$TFP. Those are two different variables in the same way as GDP and growth rate of GDP are not the same.
The idea here is simple, if output (GDP) grows, but all factors remain same, that means that the productivity of all the factors used must have increased.  However, in real life things usually don't remain constant so you will observe that quantity of inputs economy uses increases over time, but if output increases even faster that has the same implication as the case when inputs remain constant and output grows, namely it means productivity of all those factors is increasing.


What's the logic behind dividing rental price of capital and wage rate by price level? I've just started to learn macroeconomics and I think even the teachers can't get what's the price level and why we divide by it in order to find for example real economic profits.
Textbook says "What firms and households care about are profits in terms of what they can
buy, that is, real economic profits. We divide the preceding expression by the price level,
P, to get real economic profits     $$Π= F(K,L) - (R/P)K - (W/P)L$$
I just cant understand the part of dividing by P, everything else is clear.
Thank you.

Price level is the general level of prices in an economy. It is a variable that indicates what is the purchasing power of money (see Blanchard et al. Macroeconomics: a European Perspective). You can think of it as a variable that tells you what average prices are compared to some baseline.  Price level is often measured using the consumer price index (CPI).
As your textbook correctly stated economic agents primarily care about real profits, wages etc. The reasoning behind this is that what matters is not what is the number of zeros on your paycheck but what you can buy with it. For example, suppose that you are offered a job that pays ${\\\$}1,000,000$ salary per month. Is that salary high or low? Well if the average price level $(P)$ is so high, due to inflation, that even apple or postage stamp cost ${\\\$}1,000,000,000$ you would probably not be willing to work for that one million per month, because it would not be enough to even buy you one apple. However, if price level would be such that stamps and apples cost only ${\\\$1}$ that kind of salary would be great because you can purchase whole host of apples with it.
What really matters for decisions is how much goods and services you can buy with profit an wage you get. Consequently, in economics we often divide nominal variables by price levels in order to adjust them for any effects of inflation.


What conditions characterise "cheap-talk" games I've looked all over the internet but couldn't find a simple answer. My claim is that if none of the players utility functions depend on the messages being sent, then the game is a cheap talk. Consider this :

  Suppose that after you graduate you have two options when applying for a position at a firm: You can either use your undergraduate diploma or pursue graduate studies for two years and receive an MA diploma and send it as part of your application. Your skill level is either High or Low. You learn your skill level but the employer does not know your skill level. The employer only knows that your it is High with probability 1/4. After you learn your type you either apply for the job with your undergraduate diploma or get an MA degree and apply with master’s diploma. Thus you have two possible messages. You already have a BA degree, so its cost is zero. The cost of MA degree is 2 if you are High type and 5 if you are Low type. The employer, observing the message you send but not your type, decides what kind of job to offer you. There are two possible jobs: job1 pays 10 and job2 pays 6. The net payoff of the employer is 10 if you are High type and employed at job1, 5 if you are High type and employed at job2, 0 if you are Low type and employed at job1, and 3 if you are Low type and employed at job2. Is this game a cheap talk game?

We can clearly see that for the employee the messages (whether choosing a MA or BA degree) will have an effect on her payoffs. Therefore I conclude that the game is not cheap talk. But I'm not sure since this kind of games are characterised under cheap talk games, what makes a game cheap talk? How I can explain whether the game above is a cheap talk or not

Your general question is in my opinion far more interesting than the course problem, so I will answer that.

What conditions characterise “cheap-talk” games?

As you have noted yourself, the talk needs to be cheap, meaning signals/messages should have no direct effect on the players' utility. Indirect effects are allowed. Even in cheap talk games it is possible that the beliefs of a player are updated by a cheap message, hence his response changes altering the outcome. Simple examples include coordination problems.
A second necessary condition is that all messages are available to all types. hence there are no type specific message spaces $M_i$, only a shared message space $M$ from which all types of the sender can freely choose their message. Without this, a signal, though costless, could carry information, as certain types are incapable of sending it.

It's a cheap-talk game, so what?

If these two conditions are met then the game is a cheap-talk game. Why is that important?

Let us call a version of the cheap-talk game where we leave out the signal round the simple variant. For any equilibrium of the simple variant there is a payoff equivalent equilibrium of the cheap-talk game. Similarly, for any
equilibrium of the cheap-talk game there is a payoff equivalent equilibrium of the simple variant. Hence it seems that the ability to signal did not fundamentally alter the set of equilibrium outcomes. The reason for this is that the signal did not carry any information, as it was easily imitable by other types, it was 'cheap'.


Standard Macro Model Intuition In the standard model of Aggregate Demand and Aggregate Supply which display equilibrium in all the markets, I often get confused on how a change in one variable will shift curves. Say, output decreases - how can one determine the causality between a change in output with other variables?

For example, can prices change output and output change prices? What about other variables - GDP, interest rate?

Rule of thumb: if the changed quantity does not appear on an axis of the graph, its effect (if any) will be a shift of the curve, rather than movement along the curve.

For example, an AD-AS graph has price and quantity on the axes. Interest rate does not appear on the axes, so an increase in the interest rate (say) could be associated with a decrease in the money supply, resulting in a decrease in demand that shows up as a shift of the aggregate demand curve to the left.


Prove this indirect utility function is quasi-convex The indirect utility function is as follows:

$$ v(m,p) = \frac{m}{p_{1}^{1/2} p_{2}^{1/4} p_{3}^{1/4}} $$

I need to prove that it is quasi-convex. I tried both definition of a quasiconvex function and the property of a convex lower contour set but couldn't get it.

This is a question in my homework so you may just want to give hint.

If the other two methods did not work out for you, another possibility, albeit a more difficult one:

Set up the bordered Hessian:

$$\bar{H}=\begin{bmatrix}0 & \frac{\partial v}{\partial p_1} & \frac{\partial v}{\partial p_2} & \frac{\partial v}{\partial p_3} & \frac{\partial v}{\partial m}\\
\frac{\partial v}{\partial p_1} & \frac{\partial^2 v}{\partial p_1^2} & \frac{\partial^2 v}{\partial p_1\partial p_2} & \frac{\partial^2 v}{\partial p_1\partial p_3} & \frac{\partial^2 v}{\partial p_1\partial m}\\
\frac{\partial v}{\partial p_2} & \frac{\partial^2 v}{\partial p_2 \partial p_1} & \frac{\partial^2 v}{\partial p_2^2} & \frac{\partial^2 v}{\partial p_2\partial p_3} & \frac{\partial^2 v}{\partial p_2\partial m}\\
\frac{\partial v}{\partial p_3} & \frac{\partial^2 v}{\partial p_3 \partial p_1} & \frac{\partial^2 v}{\partial p_3 \partial p_2} & \frac{\partial^2 v}{\partial p_3^2} & \frac{\partial^2 v}{\partial p_3\partial m}\\
\frac{\partial v}{\partial m} & \frac{\partial^2 v}{\partial m \partial p_1} & \frac{\partial^2 v}{\partial m \partial p_2} & \frac{\partial^2 v}{\partial m \partial p_3} & \frac{\partial^2 v}{\partial m^2}\\
\end{bmatrix}$$

Define $\bar{H}_k$ as the $k_{th}$ order leading principal submatrix. We can apply the following theorem:

If the $\text{Det}\left(\bar{H}_k\right) <0$ for all $k=2,3,4,5$, then $v$ is quasiconvex in $(\textbf{p},m)$.

This is an application of the more general result:

Let $X$ be an open convex subset of $\mathbb{R}^n$ and let $f:X \rightarrow \mathbb{R}$ be a twice continuously differentiable function $\left(f\in C^2\right)$. Let $\bar{H}$ be the bordered Hessian and let $\bar{H}_k$ be the $k_{th}$ order leading principal submatrix. If the $\text{Det}\left(\bar{H}_k\right) <0$ for all $k=2,3,\dots,n+1$, then $f$ is quasiconvex on $X$.


falling household savings While reading Mazzucato's "The value of everything", I come across this

  The rise in private debt in the US and UK has resulted in household
  savings falling as a percentage of disposable income - income minus
  taxes - especially in periods of sustained economic growth (during the
  1980s, the late 1990s and the beginning of the 2000s.

I wonder if this is based on observation or if there is a cause and effect (as suggested by "resulted"), in which case I'd be grateful if anyone could provide some intuition of why such a relationship would exist.

If household debt rises, it means that more households consume more than they earn. This is the same as saying they save less. This means that the causal channel in your quote is by definition.

Now one may ask why do savings decrease in a period of growth. I could imagine two causal channels. It could be that growth is a consequence of low interest rates, causing both higher private debt and hence more investment. One could also think of sustainable growth causing positive expectations, so households are worrying less about the future and hence save less for precautionary reasons.


How much to earn to break-even throughout life Firstly, my apologies if this isn't the right forum for this - I couldn't spot a better one!

We are all born, pay taxes and die.

Depending on where we live, those taxes will (hopefully) be used to pay for our medical care when we are born, our education, maybe our college fees, health care during our lives and perhaps social care when we get too old to look after ourselves. Not to mention defending our country (armed forces) and keeping the streets safe (police).

Some of us will be fortunate enough to earn lots of money, pay tax on that money and still be able to pay for their own healthcare, their children's education and their own care when they get old. These people are likely to pay much more in taxes than they will ever receive back in benefits from the state.

Others won't be so fortunate, maybe they'll have periods of unemployment, maybe they have long-term (or just expensive) medical problems, or maybe they just won't pay much tax because they don't earn much in the first place.

In summary - some people will, over the course of their life, contribute far more in taxes than they will receive back in benefits. Others will receive much more in benefits than they ever contribute.

There must be a break-even income where a persons' overall, lifetime, tax contribution matches the total of all the benefits he receives.

So, assuming a typical working life of 45-50 years and an average life expectancy what should my average salary be over that period to eventual break-even? to be neither a net-contributor nor a net-consumer?

For the purposes of this question, I'm assuming UK rates of taxation and benefits although if there are figures for other countries I'd be happy to have those too! I'm also just asking about financial contributions/costs, not considering the relative social contributions of different types of jobs (i.e. nurses compared to stock-brokers etc)

UPDATE - I think I've found most of the answer at this link .

Generally, only the top 1% pay significant taxes. Nearly half of taxes in the US are paid by the top 1% of taxpayers.

So in order to pay an average tax burden you need to make hundreds of K a year.


Constrained optimisation with transfer I have been stuck on this question for about two days and can find no way out (apologies if the question seems really simple as I haven't started university yet). I would strongly prefer it if this can be solved using the Lagrange multiplier. Thanks.

There are two individuals, A and B, in an economy. Each derives utility from his
consumption, C, and the fraction of his time spent on leisure, l, according to the
utility function:
U = ln(C) + ln(l)
However, A is made very unhappy if B’s consumption falls below 1 unit, and he
makes a transfer, G, to ensure that it does not. B has no concern for A. A faces a
wage rate of 10 per period, and B a wage rate of 1 per period.
(a) For what fraction of the time does each work, and how large is the transfer G?

I'll guess that, since you're not yet in college but you're doing something this "advanced," you're not doing a homework problem, and that you're asking for a hint.

Since B doesn't care about A, you should be able to find B's consumption $(C_B, l_B)$ using standard method (maximizing utility w.r.t. budget constraint, etc.)

Now there are two cases.

$C_B \geq 1$. Then what would the problem for A turn out to be?
$C_B < 1$. Then think about how much should A transfer to B? Should he transfer nothing? 0.1? Up until what point would the increase in the transfer stop making A better off?


VAR inversion - looking for a good resource I am having trouble with something that should be pretty basic.

I need to invert a VAR (vector autoregression). Everything I have read just brushes past the actual inversion process, taking for granted that I already know how to invert a VAR. Does anyone know a good source (book, journal article, website) that walks through inverting? I've looked at Diebold's Elements of Forecasting and Enders' Applied Economic Times Series, as well as other sources with good coverage of VARs, but I can't find a good explanation of the basics of working with VARs.

Thanks!

To get Quah and Vahey's method, you should probably get a good handle on the Wold Representation Theorem first. Christiano has good notes on it here. Then, you can apply it to VAR. There's a pretty good basic explanation in Greene's Econometric Analysis. In the Sixth Edition, it's in Chapter 20 (Models with Lagged Variables) starting with Section 20.6, "Vector Autoregressions." It should be pretty straight forward to apply the Wold Theorem to VARs with Greene's notation. If it's still not quite clear to you, there's a pretty close example on page 704, subsection 20.6.8.b, "The Sacrifice Ratio." The basic idea is that you just end up sort of inverting the matrix of lag operators and coefficients (the "inversion" part) so that you have your main variable on the left side and a lag polynomial in your errors on the right side.


Is there a name for this generalization of the stable marriage problem? I'm thinking of the following problem: There's a set $M$ of men and a set $W$ of women; furthermore there's a set $S$ of "possible marriages". Each possible marriage is a triple $(m, w, x)$, consisting of a man $m \in M$, a woman $w \in W$ and some arbitrary piece of extra information $x$.

For a man $m$, let $S_m = \{(m', w', x) \in S \mid m = m'\}$ be the set of possible marriages involving him, and for any woman $w$, let $S_w = \{(m', w', x) \in S \mid w = w'\}$ be the set of possible marriages involving her.

Instead of ranking the women he would marry, a man $m$ ranks the possible marriages: There is a total ordering $\leq_m$ defined on $S_m$. Similarly, for any woman $w$, there is a total ordering $\leq_w$ defined on $S_w$.

The problem is to find a stable matching. A stable matching is now a subset $X \subseteq S$ such that

 every person is matched at most once, that is, for each man $m$, the intersection $X \cap S_m$ has at most one element, and for each woman $w$, the intersection $X \cap S_w$ has at most one element, and
 $X$ is stable in the sense that for each possible marriage $s \in S$ between a man $m$ and a woman $w$, either

 $m$ is in a marriage that is at least as good for him as $s$, that is, there is some $s' \in X \cap S_m$ such that $s' \geq_m s$, or
 $w$ is in a marriage that is at least as good for her as $s$, that is, there is some $s' \in X \cap S_w$ such that $s' \geq_w s$.

The classical stable marriage problem is equivalent to the restricted version of this problem where for each possibe marriage $(m, w, x) \in S$, the "extra information" $x$ is required to be some fixed dummy value.

Has anybody studied this generalization and given it some name? If not, is there some naming convention for "this kind of generalization"?

Yes, this has been studied. Look up "matching with contracts" and the earlier papers by Crawford & Knoer (1981), and Kelso & Crawford (1982) in Econometrica.


Who owns state bonds? Let's take as an example the province of Quebec in Canada with a debt of $273B. It just elected a conservative and nationalistic government
to lead the next 4 years. Their platform is that of fiscal prudence and tax reduction. There is a lot of debate around the need to lower the state's debt versus investing more into services and increasing said debt.

With that in mind, I have a few questions for you all:

Who owns all this debt? What happens when the state falls into the red? Is it possible for the state to no longer be able to borrow? Is there a finite amount it can borrow?

I’ll offer brief answers to the sub-questions.

“Who owns the debt?” You might be able to find information in Canada’s financial accounts data, but all provinces are likely aggregated. Finance Quebec may voluntarily post some information on holders, but I am unaware of any sources in the public domain.

“What happens when a province falls in the red?” Canadian provincial finance is very different than American state finance. If “fall in the red” means “runs a fiscal deficit,” the province will just issue bonds.

“Is it possible for a province to no longer borrow?” Unlike American states, there are no balanced budget laws in Canadian provinces. The “limit” is the inability to float bonds in the bond markets. The Social Credit government in Alberta was forced to default in the Great Depression (1930s), and I believe that was the only post-Confederation (1867) provincial default. (It’s hard to find an authoritative reference for that statement, sorry.)

“Is there a finite anount it can borrow?” The only limit is what it can get from auctioning bonds to investors. There is no iron law determining an upper limit for that, but realistically debt levels will be compared to past history and the situation of other provinces.

A single province is relatively small when compared to aggregate borrowing/lending in the Canadian dollar financial system. Therefore, a province is unlikely to hit some aggregate “limit,” rather it is a question of whether investors are willing to allocate their portfolios towards the province. For example, if investors fear default, they will not be happy to lend, no matter how much balance sheet capacity they have to buy bonds. Discussing the aggregate lending capacity is controversial, and I am deliberately ducking that question. That should be a separate question.


How does hyperinflation affect investment? Title says it all - how does hyperinflation affect investment and how does this differ from how creeping inflation might affect investment?

Hyperinflation has great negative impact on the entire economy, as it makes it very difficult for households and businesses to plan long term projects and protect their wealth and earning power, reducing the real growth of the economy.

Inflation increases the nominal yield of investment securities and erodes the value of a currency. You can get more information on inflation effects on investment from this PIMCO primer.


Using an elasticity to evaluate changes in a dependent variable Often in textbooks and online I see thte definition of an elasticity to be as follows:

  1) $\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\partial \log Y}{\partial \log X}$

Thus, when using an elasticity along with a percentage change in X to estimate a percentage change in Y, it would seem to make sense that one would multiply the percentage change in X by the elasticity as below:

  2) ${\frac{\Delta Y}{Y}}=\epsilon\cdot(\frac{\Delta X}{X})$

However, in one instance I've seen mention that it is more accurate to estimate the percentage change in Y as follows:

  3) ${\frac{\Delta Y}{Y}}=(\frac{X_2}{X_1})^\epsilon-1$

I'm seeking to apply elasticities derived by estimating log-log regression models and would appreciate some guidance as to whether formula #2 or #3 is more appropriate when evaluating impacts on Y resulting from changes in X.

Can someone please explain why formula #3 may be more appropriate than formula #2? And how does one get mathematically from formula #1 to formula #3?

why formula #3 may be more appropriate than formula #2?

Actually, #2 is a linear approximation of #3.
Note that
$\partial \log Z$
can be written/seen as
$\log Z_2 - \log Z_1 = \log\frac{Z_2}{Z_1} (\approx 0)$
where the above difference is infinitesimal, hence the use of $\partial$.
In such a case, #1 can be written as
$\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$
Equivalently, the above equality is only true because we are dealing with infinitesimal changes. Which means that if, say, $Y_2 >> Y_1$, the above equality implies an approximation, i.e.
$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)} \approx \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}$
Which leads to #2:
$\frac{\Delta Y}{Y} \approx \epsilon \frac{\Delta X}{X}$
Actually, the smaller $\Delta Y$ and $\Delta X$, the better the approximation.

how does one get mathematically from formula #1 to formula #3?

$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$
$\Leftrightarrow \log \frac{Y_2}{Y_1} = \epsilon \log \frac{X_2}{X_1} =$
$\log\left[\left(\frac{X_2}{X_1}\right)^\epsilon\right]$
$\Leftrightarrow \frac{Y_2}{Y_1} = \left(\frac{X_2}{X_1}\right)^\epsilon$
$\Leftrightarrow \frac{Y_2}{Y_1} -1 = \frac{Y_2 - Y_1}{Y_1} = \frac{\Delta Y}{Y}= \left(\frac{X_2}{X_1}\right)^\epsilon -1$


Federal Reserve Notes vs Coins More of a two part question. In the current system (Fiat only currency), why is a Federal Reserve Note booked as a liability (on the Fed balance sheet), but a Coin is not booked as a liability by the Treasury? What is the difference? What liability does the Fed have since it doesn't actually have to redeem the notes for anything (other than another note?)

Also, what does it mean to say that the notes are 'obligations' of the US Government?

The referenced link from money.stackexchange.com with the top accepted answer is incorrect.  It states:

"Coins are assets because its the actual money. Notes are liabilities because the Federal Reserve is obligated to pay money on these notes."

The Federal reserve is not obligated to pay money on the monetary base it creates and there is no objective financial difference between coins being real money and federal reserve notes being real money.  They are both real money for all practical purposes.

Let me take a stab at the initial question.

Yes, Federal Reserve Notes are a liability to the Federal Reserve (as are electronic federal reserve dollars).  It doesn't really need to be done like this.  It's actually a holdover from when the US was on a partial gold standard.  The central bank would hold gold deposits and in turn issue notes (dollar bills) which were redemption slips for those gold deposits.  Because the central bank had to redeem dollar bills with gold, this is why it was classified as a liability.  The US went off the gold standard twice (once under FDR to the public and later under Nixon for foriegners).  When this happened, dollars were no longer true liabilities.  But this history is why you see federal reserve notes (and federal reserve deposits which are electronic dollars) still classified as liabilities.  And coinage is considered an asset to the Fed which they purchase from the treasury.  You will still see some gold as an asset on the Fed balancesheet as a holdover from these times.

The Federal Reserve does not have a monopoly on creating the monetary base.  The treasury can also create base money by creating coins and US Notes (which are not Federal Reserve notes).

The treasury does not consider the coin as a liability because it truly is not one (just as the dollar bill in a modern sense is not one to the Fed).

Looking at a balance sheet, you could almost argue that base money is a liability because it appears to be balanced by assets on the balance sheet.  But this is not the case.  It is an accounting illusion. If the Fed were to withdraw all their "liabilities" there wouldn't be any money in the system (outside of coins and notes of course).

The reference to the obligation on Federal Reserve Notes is an antiquated holdover from the gold standard era.  It is quite meaningless now.  The Fed will exchange if demanded electronic dollars for paper dollars, vice versa or will buy/sell coinage as required by the banks.

Coinage is not an obligation to the treasury, neither are US notes.  The treasury will certainly repair damaged coins, but that is not to say that the coins themselves are true liabilities or obligations.


Constant Elasticity of Substitution: Special Cases Take an $n$-commodity constant elasticity of substitution utility function,

$$U = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho}$$

How do we show the following:

Show that as $\rho \rightarrow 0$ that this utility function represents preferences for Cobb-Douglass utility. $U(x) = \prod^n_{i=1} x_i^{\alpha_i}$
Show that as $\rho \rightarrow - \infty$ that this utility function has the indifference curves as in Leontief utility. $U(x) = \min\left\{x_1,...,x_n\right\}$

We know that if $u$ represents $\succeq$ on $X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then $v(x) = f(u(x))$ represents $\succeq$ on $X$

($X$ in this case is $\mathbb{R^n}$)

Consider $v(x, \rho) = \ln(u(x, \rho)) - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho}$, which is strictly increasing.

$$v(x, \rho) = \frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right]}{\rho} - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho} =
\frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right] - \left [\ln\sum^n_{i=1}\alpha_i \right]}{\rho}$$

The limit of this as $\rho \rightarrow 0$ is indeterminate, $\frac{0}{0}$. So we can use L'Hopital's Rule and take the derivative with respect to $\rho$ of the numerator and denominator.

$$\lim_{\rho \rightarrow 0} \frac{\ln\left[\sum^n_{i=1} \alpha_i x^\rho_i \right] - \left [\ln\sum^n_{i=1}\alpha_i \right]}{\rho} =
\lim_{\rho \rightarrow 0} \frac{1}{\sum^n_{i=1} \alpha_i x_i^\rho} \cdot \left(\sum^n_{i=1} \alpha_i x_i^\rho \ln x_i\right)$$

by the Chain Rule.

$$= \lim_{\rho \rightarrow 0} \frac{\sum^n_{i=1} \alpha_i x_i^\rho \ln x_i}{\sum^n_{i=1} \alpha_i x_i^\rho} = \frac{\sum^n_{i=1} \alpha_i \ln x_i}{\sum^n_{i=1} \alpha_i} = \frac{1}{\sum^n_{i=1} \alpha_i} \cdot \ln\left(\prod^n_{i=1} x_i^{\alpha_i}\right)$$

Consider $w(x, \rho) = \mathrm{e}^{(\sum^n_{i=1} \alpha_i) \cdot v(x, \rho)}$, which is another monotonic transformation, strictly increasing. So $w$ still represents the same preference as $u$.

$$\lim_{\rho \rightarrow 0} w(x, \rho) = \mathrm{e}^{(\sum^n_{i=1} \alpha_i) \cdot \lim_{\rho \rightarrow 0} v(x, \rho)} = \prod^n_{i=1} x_i^{\alpha_i}$$

which is a Cobb-Douglas function.

$\square$

To show the second point, it is sufficient to show that

$$\lim_{\rho \rightarrow -\infty} u(x) = \left\{x_k \ \forall j \neq k \mid x_j \geq x_k \right\}$$

$$u(x) = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho} = x_k \left[(\sum^n_{i=1, i \neq k} \alpha_i x^\rho_i) + \alpha_k \right]^\frac{1}{\rho}$$

$(\frac{x_j}{x_k})^\rho \rightarrow 0$ as $\rho \rightarrow -\infty$ if $x_j > x_k$

$(\frac{x_j}{x_k})^\rho \rightarrow 1$ as $\rho \rightarrow -\infty$ if $x_j = x_k$

So

$$\lim_{\rho \rightarrow -\infty} x_k \left[(\sum^n_{i=1, i \neq k} \alpha_i x^\rho_i) + \alpha_k \right]^\frac{1}{\rho} = x_k$$

since $1/\rho \rightarrow 0$ and a constant to the zeroth power is 1.

Construct a similar argument for any $k$. Thus $\lim_{\rho \rightarrow -\infty} u(x) = \min \left\{x_1,...,x_n \right\} $

$\square$