Implications of declining money velocity In the United States, the velocity of both the M1 and M2 money supplies has been declining for over a decade.

The decline in the velocity of M1 could be explained by the shift away from cash to cards and online payments.

What are the causes and implications of the decline in the velocity of M2?

Does it signify that the money supply is growing faster than GDP? If so, is real GDP declining? Does it indicate some sort of lack of liquidity? How does this relate to interest rates?

Thanks!

http://www.forbes.com/sites/traceygreenstein/2011/09/20/the-feds-16-trillion-bailouts-under-reported/#5a1dec1d6877

The decline in money velocity is due to enormous monetary injections, which mostly sat in recipient bank accounts and had no effect on the real economy. Under the Keynesian and monetarist theories that central banks operate upon, an increase in money supply should ease liquidity constraints and allow the economy to resume growth. In reality not much further growth was possible, likely because of the stagnation in conventional oil production.

To answer your questions:

Yes
Not yet
No, it indicates a lack of viable uses for the cash
Interest rates fell because of the relatively small decrease in banks balance sheets coinciding with a sharply declining return on capital, and thus diminishing incentive for investors to leverage, causing increasing supply relative to demand for credit


What's unique about the Chicago School's Methodology? What is the methodology of the Chicago School and what makes it unique among economic schools of thought?

I would submit that the premise is flawed. There is no methodological criteria that unified the Chicago school and separates it from the rest of economics. I think this is true even for Friedman, whose methodological paper you refer to. If you look at his macroeconomic work such as his Monetary History of the United States, Theory of the Consumption Function or his famous presidential address to the AEA,there is nothing in it which is different from the methodology of mainstream economics. I think the same can be said about Stigler's work on industrial organization.

The Chicago School was united by its institutional links and its faith in capitalism and not by a distinctive method.

And it would be wrong to think of the School as a homogenous entity. See for example this article by Patinkin on the diversity of views on the quantity theory of money:

http://stevereads.com/papers_to_read/the_chicago_tradition_the_quantity_theory_and_friedman.pdf


How to find corner Pareto efficient allocations I have some troubles in understanding how to find Pareto efficient allocation that are on the frontier of the Edgeworth box. I mean, the interior ones, can be found using the equality $MRS_A = MRS_B$, where $A$ and $B$ are the two agents of our pure exchange economy, whereas I cannot find anything about frontier allocations.

Is there any mathematical procedure I can use or it is only something I have to deal with using intuition (using the definition of Pareto efficiency I mean).

To be clearer, I cite an exercise from my workbook where there are two agents with preferences $u^A(x_1,x_2) =\sqrt{x_1x_2}$ and $u^B(x_1,x_2)=x_1+3x_2$, and endowments $\omega^A=(1,0)$ and $\omega^B=(0,1)$. The question is to identify the Pareto set.

This is the solution, where the blue lines are the Pareto set.

Let us take your example:

First, we note that both utility functions are differentiable and quasi-concave.
Noting this, we also know that the necessary and sufficient condition for internal Pareto optimality is that $MRS_{x_1,y_1}$=$MRS_{x_2,y_2}$ (as you have already correctly stated).

This condition will clearly coincide with the portion of the solution identifying the locus on internal P.O. allocations.

Now, for the P.O. points along the right edge:

We can find the bound of internal solutions by identifying the range over which the MRS condition noted above fails.
Because the equality fails, we know a strict inequality must prevail.
The directionality of the prevailing strict inequality identifies the edge along which we find our P.O. allocations.

So, there are two ways to answer your question, I think.

1.) For this type of graph, where one agent has linear preferences and the other has curvilinear and convex preferences, it is easy to see that the locus of P.O. allocations shifts toward the right edge of the Edgeworth box. Thus, corner solutions run along that edge across the range where points of tangency are no longer interior.

2.) If you encounter a situation where, for example, both agents have linear preferences, you can use the directionality of the strict inequality to identify along which edges of the Edgeworth box you have P.O. allocations.

An example for 2.) :

$U_1(x_1,y_1)=X_1+2Y_1$ and $U_2 = 2X_1+Y_1$

Now, you should do the following:

ensure you understand why $MRS_1 \neq MRS_2$
determine the direction of the strict inequality between the two
use this to identify along which edge of the box you will have P.O. allocations.

Hints:

$MRS_A<MRS_B$
the P.O. allocations are along the left and upper edges of the Edgeworth box

Edit:

I also think this is a good reference:

http://nicolalimodio.com/wp-content/uploads/2015/11/ec202d.pdf

Hope something in there helps. This was an issue for me as well when I began learning such things and it took me a bit of reading and practice to finally become comfortable with it all.


How can governments seemingly run permanent budget deficits? How can countries afford to run budget deficits for decades on end and not become overburdened by interest payments on those debts?

Because they have low interest rates and growing tax revenue.

As long as your income is growing faster than your debt, you can keep borrowing.


Identifying utility function I recently came across a utility function with min written at the start. I assumed that it was a case of a leontief utility function, and only after going ahead with the problem I found out that it is a case of perfect substitutes. Until then, I didn't even know that perfect substitutes could take this form. How do I identify various utility functions accurately? Are there other such confusing cases which I need to be aware of?

The example I have mentioned above looks something like this:
      U = Min [2x + y, x + 2y]

Not sure about utility function being $\min$ and it being a case of perfect substitutes...

In general, I would say there are a few different types of utility functions but if you know more or less the theory and can imagine various functions' graphs, then (as mentioned in the comment) you will be fine.

So I think we should start with a Cobb-Douglas preferences which can be represented by utility function of the form: $$u(x_1,x_2) = x_1^ax_2^b \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Then you have a perfect-substitute preferences which can be represented by utility function of the form: $$u(x_1,x_2) = ax_1 + bx_2 \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

You can also have a perfect-complement preferences which can be represented by utility function of the form: $$u(x_1,x_2) = \min\{ax_1,bx_2\} \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Finally, you can have a quasi-linear preferences which can be represented by utility function of the form: $$u(x_1,x_2) = f(x_1) +x_2 \:\:\: \textrm{where f is increasing and concave}$$

In all of the above graphs I assumed $a=b=1$ for simplicity. In the last case the concave function was a square root.
Your example is (more or less):

You can now try and argue what kind of preferences these are.

If you are interested in consumer theory, you should find some proper course book or other materials - there's plenty of it on the Internet - just google it.

The last piece of advice I can give you - try to draw as many graphs as you can think of and then try to find a story that the graph is explaining. Or the other way round. I remember my professors trying to "play" with us by giving us different kinds of utility functions - when you drew them, they looked "strange" but when you actually thought about them - they did have some story to tell.

EDIT.
There is also a special case in a way - monotonic transformation is a real-valued function $f: \mathbb{R} \to \mathbb{R}$ which basically transforms numbers into numbers in a way that preserves the order of the numbers, that is, for every two numbers $p$ and $k$ it hods that $p>k \implies f(p)>f(k) $. This means that if a function $f$ is a monotonic transformation of utility $u( \cdot )$, then $f(u(\cdot))$ also represents the consumer’s preferences. $$(x_1,x_2) \succeq (y_1,y_2) \iff f(u(x_1,x_2)) > f (u(y_1,y_2))$$

All this means is that you need to be careful for a case of utility function like: $\ln{x_1} + \ln{x_2}$ which is just a natural logarithm of Cobb-Douglas preferences $x_1x_2$ with $a=b=1$.


What are examples for the phenomenon that more (or better) information makes everybody worse off? More information is usually considered "better". Let's say a rational agent chooses optimally given his information on the circumstances of a particular decision problem. Then providing him with more or better information makes him (weakly) better off.

Other agents might be made worse off in the process, and total welfare might even decrease. Danove et al. (2003) e.g. argue that the introduction of mandatory "report cards" in New York and Pennsylvania in the 1990ies to evaluate bypass surgeries reduced welfare because it incentivized doctors to avoid treating very old or sick patients. In this example all patients got more information, and some of them, but not all, ended up worse off.

But there are also circumstances where providing agents with more information makes everybody worse off. When visitors are about to leave a crowded theater, truthfully informing them that a fire just broke out behind the curtain might have disastrous consequences for all of them. Informing the public that a particular bank is in trouble could trigger a bank run that leaves everybody worse off.

What are other such phenomena? I am looking for examples (in theoretical models or in the real world) of the latter kind, where exogenously providing a group of agents with more or better information leads to a new equilibrium which is Pareto inferior compared to the old one.

Traffic reports are another type of information that potentially makes everyone worse off.

A recent working paper (Wiseman and Wiseman 2019, https://drive.google.com/open?id=10B6VudYJOQB5w2UVBrYEjYc5wwzGZc8Z) shows this in a simple model of traffic.

I quote from the paper, "...public information about road conditions tends to increase average travel time when...3) the scale of the traffic network is large." This corresponds to their Theorem 3, where "...in a scaled-up version of any environment with excess capacity, traffic reports increase expected travel time."

Yet another broad way of thinking about your question is in terms of monitoring structures of repeated games. A recent paper (Sugaya and Wolitzky 2018 JPE, http://economics.mit.edu/files/11455) look at imperfect monitoring in cartels and show that less transparency can facilitate collusion, making all participants better off.


What's a more accurate term than 'share price' that doesn't depend on (reverse) stock splits? Purport that retail investors desire to invest in AAPL or TSLA (both have been reported to split their stock soon), because they think their "future share price" (after the stock split) will skyrocket. Is there a more accurate term for "future share price"?  "share price" feels inaccurate if you are disregarding (reverse) stock splits.
Is it "market capitalization"? I know that market capitalization signifies "the total value of all a company's shares of stock. It is calculated by multiplying the price of a stock by its total number of outstanding shares."
To wit, at the moment when stocks split their stock, their "share price" will drop, but their [fill in the blank] ought stay the same.

How about "value"? Market capitalization seems fine too.


Beta as an exponent? Can beta be an exponent within a function prior to taking the log of that function to arrive at a simple, log-linear estimation equation?

I am curious because I found an equation that had beta as an exponent on page 7 of this article by Lee Branstetter(2004): http://bit.ly/1OGSFx9 titled: Is Foreign Direct Investment a Channel of Knowledge Spillovers? Evidence from Japan’s FDI in the United States

I believe there this is not out-of-order. I am however, seeking an explanation to the equation.

I have tried to check different sources and the closest explanation is that the function is a linear-log function.

This is not a structural paper. The authors are not "deriving" a log-linear relationship from a model, which they then proceed to estimate. They rather start by postulat the relationship:

  Let $C^J_{it}$ be the number of citations made by the patent
  applications Japanese firm i filed in year t to the cumulated stock of
  bindigenousQ U.S.-invented patents granted as of year $t$. I can then
  write the log of $C^J_{it}$ as a simple log-linear function of several
  other observables

  $$ c^J_{it} = \beta_0 + \beta_1 p_{it} + \beta_2 FDI_{it} + \gamma_i +
> \alpha_t + \epsilon_{it}$$

which is their equation (1).


Thin indifference curves If a consumer follows the rationality axiom of continuity (i.e. no jumps in his preferences), the indifference curves of a utility function are said to be thin.

Why does continuity ($x \succeq y \Rightarrow \exists \space z=x+\epsilon$ such that $|z|\ge y \space \forall \epsilon > 0$) imply thin indifference curves?

To begin with, I think the question is wrongly stated. For if the defininition of a thin indifference curve is such that continuity of a consumer's preferences implies thin indifference curves, then, surely, continuity implies thin indifference curves... This answers your question.

However, if we are to make a suitable definition of a thin indifference curve, we may firstly say that $$[q]=\{p\in\Delta|p\sim q\}$$ is a thick indifference curve, where $\Delta$ is the set of possible bundles, and where $\sim$ denotes indifference, whenever there exists a $q'\in [q]$ and a $\epsilon>0$ such that $p\in N_{\epsilon}(q')$ implies $p\sim q'$, where $N_{\epsilon}(q')$ is some epsilon-neighbourhood around $q'$; and secondly say that $[q]$ is a thin indifference curve if it is not thick. Colloquially this means that there is some bump on a thick indifference curve $[q]$, but no such bump on a thin indifference curve.

Essentially, the above is a short exposition of A Geometric Approach to Expected Utility (Chatterjee & Krishna, 2006). Using the above definition of a thin indifference curve, they show in Lemma 2.3 that (i) continuity and (ii) independence implies thin indifference curves (note that they do not show that continuity alone implies thin indifference curves; cf. Ubiquitous' answer). Their definition relies on the following two topological concepts.

The assumption of continuity. All subsets $\{q|q\succ p\}$ and $\{q|p\succ q\}$ of $\Delta$, where $p\in \Delta$, are open; here, remember that an open set is a set for which each point in it has a neighbourhood lying in that set. Thus, this notion of continuity is similar to yours.
The assumption of independence. For all $p,q,r\in\Delta$, $p\succ q$ and $\lambda\in (0,1]$ implies that
$$\lambda p+(1-\lambda)r\succ \lambda
q+(1-\lambda)r;$$ this allows for some nice algebra.

Now, what they show in Lemma 2.3 is essentially that if you have an indifference curve $[q]$ and consider some epsilon-neighbourhood $N_{\epsilon}(q')$ around $q'\in [q]$, then $p\in N_{\epsilon}(q')$ will not imply that $p\sim q'$ for arbitrarily small $\epsilon>0$. I.e., however small, no epsilon-neighbourhood is such that it only contains bundles for which one is indifferent between those bundles and $q'$. Instead, every epsilon-neighborhood will include points that are strictly preferred to $q'$.

For continuous utility functions, I think it is fruitful to note that their image in e.g. $\mathbb{R}^2$ has (Lebesgue) measure 0 (cf. How to prove that the image of a continuous curve in $\mathbb{R}^2$ has measure $0$?)


because bitcoins do not generate earnings, then (it is a ponzi scheme)? What does the author mean with:

  Since Bitcoins do not generate any actual earnings, they must
  appreciate in value to ensure that people are willing to hold them.

Why must they appreciate in value?  Is the assumption that people must  hold bitcoin?  Why is it necessary or required to hold bitcoin?

Or, is the author saying "eh, what goes up must come down."  Instead, the article seems argue that bitcoin will go to a "value" of zero:

  ...Bitcoins are the most demonstrably valueless financial asset ever
  created...

Isn't one of the criticisms to using gold as a currency that it's then tied up as a currency and not being used functionally?  Does the inherent valuelessness of bitcoin, or any other cryptocurrency, make it suitable for transactions?

Sure, bankers hold currency, but few individuals actually hold currency directly.  Generally money is spent or invested in quick order.  Which seems not directly related to adoption of bitcoin (or other cryptocurrency) for transactions.

What is the reasoning behind the notion that because bitcoin has no intrinsic value therefore it's not suitable as currency?  Or, to put another way, if no one "held" US dollars, but simply used them to spend or invest, would that invalidate it as a currency for buying chickens?

http://nationalinterest.org/commentary/the-bitcoin-bubble-bad-hypothesis-8353?page=2

What is the reasoning behind the notion that because bitcoin has no intrinsic
  value therefore it's not suitable as currency? Or, to put another way, if no one
  "held" US dollars, but simply used them to spend or invest, would that
  invalidate it as a currency for buying chickens?

The author addresses the difference between bitcoin and fiat currency (such as greenbacks):

  Because of their power to tax, governments can make money by fiat, simply by
  declaring their willingness to accept that money in repayment of tax debts.

I'll be honest and admit that I hadn't heard that argument before and off the cuff, it makes sense.  Because the government can impose taxes and require they be paid in dollars, the demand for dollars will exist at least as long as the government has the power to tax.

I find this part of the argument to be pretty weak:

  As with any kind of asset used as currency, from gold to tobacco to U.S.
  dollars, Bitcoin is valuable as long as people are willing to accept it. But
  in all of these examples, willingness to hold the asset depends on the fact
  that it has value independent of that willingness. Tobacco can be smoked or
  chewed, gold can be used to fill teeth or make jewellery...

The problem here is that the demand of gold for jewelry and other industrial purposes is minuscule and can't come anywhere close to justifying the price of gold (even in down markets.)  If we believe this argument, the price of gold is going to go down so far that it might as well be zero.

The argument that I find convincing for why gold has been used as money for so long is that it's physical properties make it extremely suitable for that purpose.  It's not consumable (e.g. you don't put it in your pipe and smoke it.)  It is extremely stable: the list of fluids that will dissolve it is extremely short, it doesn't rust or sublimate.  It is easy to smelt.  It's heavy; making small amounts easy to measure.  It's purity is relatively easy to determine.

It created a universal standard against which the value of other things could be measured so that people didn't have to determine how many chickens was a cow worth.  Even if you were bartering, the relative value of the items could be determined based on their price in gold.  As evidence of this it is said that the discovery of the touchstone sparked the explosion of trade in the ancient world.  Nowadays very few people accept gold as payment so the demand for gold seems to be mostly fear-driven.  People believe that if fiat currencies falter or fail, gold will return to it's position as the universal measuring stick of value.  Only time will tell whether they are right.

Where does bitcoin fit into this?  It seems to me part the demand for bitcoin comes from illegal activity.  As long as hackers continue to demand ransoms in bitcoin, it's a bit like fiat currency.  It's not a measure of value.  It's value is measured in terms of fiat currencies like dollars.  Bitcoin meets a need: a digital equivalent of cash.  Technically it is traceable but it more or less fits that need.  The biggest concern for bitcoin should be is that there's nothing inherent about it that makes it unique.  Unlike gold, another competing system could be created at any time.  There's been talk of a bitcoin 'fork'.  If someone were to come up with a better system (say something that is impossible to trace) it might very well drop to zero.


Bond prices and Trade deficit Why does the narrowing of trade deficit cause an increase in bond prices?
I do understand that narrowing of trade deficit leads to narrowing of capital account surplus. Also, investor confidence rises which might push the demand for bonds high and cause an increase in bond prices?

But is there any other reason?

lower deficit means more export, so more foreign currency is coming into the country. Part of this goes to buy bonds. More demand means higher bond prices. People in the countr might feel richer too, because there is more exports, so more income to them. They might buy more bonds too.


Does the existence of insurance markets negate the broken window fallacy? In the wake Harvey and Irma, I did some reading on the Broken window fallacy. It makes sense on it's face that after paying the glazier for the broken window, the shopkeep could not spend his six francs elsewhere, so nothing of value was gained from the breaking of the window.

In the modern world though, the shopkeep probably sets some portion of his funds aside (in the form of either paying an insurer or landlord (who in turn pays his insurer)). Let's also take the child out of the equation and say the window was broken by something an insurer wouldn't deny a payout on, like a very very localized hurricane (it only broke one window, but shattered it to pieces!).

The shopkeeps six francs would not have gone to pay for shoes or bread or the like, they would have gone to the insurance company, possibly at a higher value (let's say he made no claim in a year, with 12 monthly premiums at 1 franc a piece to pay for a six franc window).

For the insurer, he may use some of these funds to pay for his own shoes, but as a responsible insurer he knows he needs to keep a few liquid francs in order to pay out for things like broken windows -- Ergo he probably wouldn't have spent the six francs either.

In this model, it seems like the glazier has produced a product and received payment to nobody else's detriment; ergo the broken window fallacy has broken.

No, they don't negate the broken-window fallacy.

You're overthinking it, and that's leading you to the wrong conclusion.

The breaking of the window represents a loss of wealth. The activities around it are neither here nor there - they do not change the fact that the breaking of the window represents a loss of wealth.

The existence of functional insurance markets just means that there are somewhat efficient markets to estimate the cost of that damage.


References on Economics of Vouchers I am interested in key references on the economics of vouchers. In particular I am interested in asking, "Under what conditions are vouchers more or less useful?"

Tangentially related resources of a high quality are also welcome.

The kinds of references which I am looking for:

Papers and books
Informal works (eg a YouTube recording of a good talk)

An overview of vouchers that is a bit outdated but a good place to start building an understanding of past work:

http://www.nber.org/papers/w7092

On school vouchers:

https://www.aeaweb.org/articles?id=10.1257/aer.97.3.789

http://faculty.smu.edu/millimet/classes/eco7321/papers/rouse.pdf

https://www.aeaweb.org/articles?id=10.1257/aer.96.3.847

https://www.aeaweb.org/articles?id=10.1257/aer.90.1.130

https://jswinton.gcsu.edu/ECON%203640/Expenditure/voucher%20potential.pdf

housing vouchers:

https://www.aeaweb.org/articles?id=10.1257/000282804322970788

https://www.aeaweb.org/articles?id=10.1257/pol.20130064

https://economics.nd.edu/assets/153486/carr_jillian_jmp.pdf

http://www.nber.org/mtopublic/482.pdf

^^^ Those are a few of the papers I've read over the last two years involving vouchers.


Solow model and a change in the savings rate In the solow model why does an increase in the savings rate shift the investment curve instead of move along it? Intuitively not mathematically
Thanks

I don't know how this can be answered without any recourse to at least basic mathematics, since the question itself refers to movement along some curves which is mathematical concept. Hence I will go over some math and then provide intuition at the end.
Explanation Including Math
The investment curve in Solow model is defined as $sf(k)$ where $f(k) = Y$ and it is assumed that when we have zero capital per effective worker $k$ output is also zero $f(0)=0$.
The curve appears to shift upwards due to change in its slope. The investment curve will have to go through the origin no matter what it's $s$  changes the slope of the investment curve. For example, if $s_1=0.5$ and $f(k)=k$ the investment curve will be: $0.5k$. If we change $s$ to $s_2=0.7$ investment curve would be now $0.7k$. This would work for any investment curve that goes through the origin and by our assumption it has to (you can try to verify this for $f(k) = \sqrt{k}$ or other valid functions.
Non Mathematical Explanation:
The investment function is by definition the proportion of output that is being saved (and hence invested). Here $s$ is the proportionality constant that determines what proportion of output is invested. Increasing that proportionality constant has to change the slope of investment function as now at any level income higher proportion of income is invested. For example, with $s_1=0.5$  if $Y=100$ the investment will be $50$, with $Y=200$, the investment will be $100$ etc. Now if you change that proportion not only the investment will be higher at any income but also rate of change of that investment will be higher. With $s_2=0.7$ at income $Y=100$, investment will be $I=70$, at $Y=200$ investment will be $I=140$.


Is there any metric to measure the creation of products and services needed to sustain and expand life? I was thinking about the value of those products and services which are needed to sustain people's lives and to expand people's lives, and how unaware I'm of a metric that measures actually this. I know GDP doesnt measure this. The inclusion in GDP of services which to me doesnt have any value in this sense, and later using GDP as a measure of a country progress doesn't make sense to me. For example, according to this

Rethinking GDP

  Yet as a statistic, it is surprisingly poor as a means of actually
  measuring what it is supposed to measure. It measures a rough
  aggregate of the total number of transactions made within a given
  country. It's used primarily because it is easy to calculate, but it
  does not really matter to this calculation whether the transactions
  involved are used when building new houses or repairing houses after a
  hurricane. Building a bomb, throwing a lavish party, or paying for
  school, it doesn't matter much what the transactions actually do, the
  statistic is only really concerned with the amount of money passed
  between hands.

So if I go to an arcade, and I spend money in it playing videogames, I'm growing GDP?

So I was wondering,

Is there any metric to measure the creation of products and services needed to sustain and expand life?

By sustaining life I mean, food, shelter, electricity, water, health care, etc. and by expanding life I mean development of technology and science in this sense.

I could understand that exporting a service would bring money to a country which later could be used to buy these things. That could make sense to be included in a metric like the one I'm looking for. But services which are pure for entertainment or comestic procedures or things in those lines doesnt make any sense to me to be included in a metric which is used to measure progress.

“So if I go to an arcade, and I spend money in it playing videogames, I'm growing GDP?”

Yes this is correct.

“Is there any metric to measure the creation of products and services needed to sustain and expand life?”

The national accounts reported by most statistical offices are often subdivided into different sectors such as agriculture, industry etc. if you would want you could exclude entertainment industry but nobody is doing that as an official separate measure because in economics we care about all humans needs not only the most basic ones like shelter etc. and entertainment is actually an important human need. However, you are free to make any measure of “progress” as you define it from those separate accounts. You can for example check gross product or gross value added by sector for European countries at Eurostat website.


Establish demand curve with infrequent sales I sell variations of an item on eBay. It is an unusual commodity which I purchase in bulk lengths and then cut to various sizes. They come in about 20 types, which I then sell as 10 different lengths, giving a total of 200 variations.

My goal is to determine what price I should charge for each, to maximise monthly profit.

The problem is I only make an average of one sale per day. I have no idea how I can establish a demand curve for the products, given such limited data. (Some variations I only get one sale per 18 months!)

What techniques could I use for this problem?

How could I go about establishing the most profitable price for each of the variations, with such limited information? (Given known costs).

The only clear idea I have is this:

Treat all 200 variations as a single product, and vary all their prices simultaneously, to derive a single demand curve. So for example, one month I might set all prices to generate 10% profit on top of the costs, then next month, set all prices to generate 20% profit, and so on, and this way establish what price generates the highest monthly profit.

But the variations tend to have wildly different applications, as well as, I expect, alternative products that can satisfy the customers' requirements, and so on - so having a single demand curve could leave significant profits untapped.

The only other idea I have is to profile the demand curve for each item in a different way:

For each product, take an example: one that only sells approx every 6 months on average: On each sale, if the time since the last sale was 6 months, I divide the total profit made on that item by 6 - or for a product that sold twice in one month, multiply by two. And then each time there is a sale I can some how establish the demand curve for it.

But the snag is for a product that sells infrequently, the demand curve could be static, but obviously this does not mean it would sell, say, once every 6 months like clockwork. So it could take years to establish a demand curve for that variation.

I'm wondering if I should entirely forget about the demand curve and concentrate instead on a marketing-oriented approach, for example:

Surveys: (How I would phrase this I'm not sure - I doubt customers would tell me the maximum they'd pay!)
Market research: plot graphs of how much other retailers/distributors charge for each variation. (Although I have seen very few indeed that sell small lengths such as those that I offer)

What you're looking for is the price elasticity of demand of your product.  That is: by how much will your sales rise or fall depending on how you manipulate price?  It depends on what you're selling and, like you said, each variation of your product will have a different elasticity.  So, say you increase prices by 5% and experience a %5 decrease in sales; your price elasticity of demand would be -1 (unit free).

The strategy you outlined to parse out the price elasticity sounds reasonable from a data collection standpoint, albeit a bit risky.  I might change the price of just one of your variations (maybe one that sells somewhat more frequently)  to assess the sensitivity your customers are to price changes.  However, if your product's elasticity is high, your sales might drop dramatically or stop altogether.


What are the defining features of Capitalism? What are the necessary and sufficient conditions for an economic system (or mode of production) to be classified as "Capitalism"?

Is it markets? Is it for-profit corporations? Is it limited liability companies? Is it workers selling their labour to capitalists/employees? Is it freedom to set up companies? Is it a low degree of government productive activity? Is it a subset of these or other elements?

I imagine the answer would depend on a particular schools of thought, so in principle I would expect different answers.

Disclaimer 1: this question is not requesting opinions about what capitalism is, it is not, it should be, it should be not, but authoritative answers, with proper references.

Disclaimer 2: for those voting to close this as too-broad, I asked on meta whether this question was on-topic, and (albeit not a popular post) it seems people agreed it was.

In this article, sociologist Erik Olin Wright provides a definition of
capitalism (from a socialist tradition):

  Many people equate capitalism with the idea of a market economy. Capitalism is not simply an economy in which markets play a central role in
  coordinating economic activity; it is a specific kind of  market  economy,  one  in  which  workers  do  not own and control the firms in which they work, capital  is  privately  owned  and  allocated  to  alternative purposes  on  the  basis  of  private  economic  returns  (or, equivalently, on the basis of private profit-making),  and  labor  is  allocated  to  economic  activities through labor markets.


Repeated game with stage game has more than 1 NE Given the game like Prisoner's dilemma (infinitely undiscounted) and each person has 3 actions in every period: confess defect and kill, and the payoff matrix have 2 NE as (D,D),(K,K).

Can I still use the property that if in any stage NE is played, then that kind of strategy is SPNE.

It seems work for me to prove that any sequence like dd,kk,kk,kk,dd....randomly combination of dd,kk, is SPNE.

Is that correct? Can I generalize that property?

Though a bit unconventional to assume an infinitely repeated game without discounting, the short answer to your question is: yes. Any sequence of stage game Nash Equilibria is supportable as a SPNE. See, for example slide 11 of this deck, or these notes from Levin, starting on page 6. This is more directly addressed by the definition of SPNEs, found on page 137, Definition 132 of these notes. Or where those Columbia notes suggest on page 142: "For example if there are multiple Nash equilibria in any of the final subgames then there are multiple sub game perfect Nash equilibria in the game itself"

Though you should note that, given your problem's construction (with no discounting), the set of SPNEs grows substantially, as you can see.


Find utility function given indifference curve? Given an indifference curve, how do you go about finding a utility function?

For example, given $z= \frac{k^\frac{1}{\delta}}{x^\frac{\alpha}{\delta}y^\frac{\beta}{\delta}}$ (defined by $U(\cdot) = k$), find a utility function. To do this, would I have to assign an arbitrary number for the utility and rewrite the function? I'm confused about what to do and can't find anything in my textbook about it. Thanks

VCG's comment about isolating $k$ is the correct approach.

Given $U(\cdot) = k$ and
$$z = \frac{k^\frac{1}{\delta}}{x^\frac{\alpha}{\delta}y^\frac{\beta}{\delta}}$$
raise each side to the $\delta$ power:
$$z^\delta = \frac{k}{x^\alpha y^\beta}$$
and isolate $k$
$$k = x^\alpha y^\beta z^\delta$$

Since $k$ is now a function of each of the goods in your equation, it makes sense as a function form for utility.

$$U(x, y, z) = x^\alpha y^\beta z^\delta$$
So we arrive at the Cobb-Douglass utility form.


Deriving the Euler equation from a Continuous Time Dynamic Programming Problem (HJB) Solving for the Euler equation in discrete time is farily straight forward with the use of the Benveniste Scheinkman theorem. However for the following standard Ramsey model:
$$\max \int_{0}^{\infty}e^{-rt} u(c(t))dt$$
subject to:
$$u(c(t))=\ln(c(t))$$
$$\dot{k}(t)=Ak^\theta-\delta k(t)-c(t)$$
the problem is simple if I use the Hamiltonian however if I write the Hamilton-Jacobi-Bellman equation for this:
$$rV(k(t))=\ln(c(t))+V'(k(t))\dot{k}$$
I have no clue how to get started. How would one go about solving for the Euler Equation using the corresponding HJB equation?

As already commented, the equation you probably meant is
$$
\rho V(k)= \sup_c \{\, u(c) + V'(k) ( f(k) -\delta k -c ) \,\}.
$$
I have never seen this equation called the HJB equation (probably missing a basic reference on my part). I'll call it "dynamic programming PDE".
What you're really asking is the connection between two approaches to solve control problems---Calculus of Variations/Optimal Control vs. Dynamic Programming.
Optimal Control (Pontryagin's Maximum Principle) is a first-order perturbation argument and Dynamic Programming Princple is a backward induction argument.
"Euler equation" arises from a first-order perturbation argument.
(In continuous-time, it's a classical Calculus of Variation equation. In discrete time, I've only heard it used in economics, describing intertemporal consumption smoothing, but it's a perturbation argument
just the same.)
In continuous-time, first-order perturbation of optimal path means a perturbation along the entire path.
In contrast, in discrete time, it suffices to perturb a single period--therefore one can obtain the Euler equation by simply differentiating the Bellman equation (under e.g. Benveniste-Scheinkman assumptions that ensure differentiability of value function).
In continuous-time, I don't believe the answer is as trivial.
There is, however, a classical connection between Optimal Control and Dynamic Programming via the method of characteristics.
Part of the connection is that, if the value function is sufficiently smooth, then the characteristic equations of the dynamic programming PDE gives
Pontryagin's Maximum Principle.
In your growth example, the maximizer of RHS is given by $u'(c^*) = V'$. Substituting then gives an implicit
first order ODE
$$
F(k, V, V') = \rho V  - u(c^*( V')) - V'\cdot( f(k) -\delta k) + V' \cdot c^*( V') = 0.
$$
Following the method of characteristics heuristicly, one of the characteristic equations would be
\begin{align*}
\frac{d}{dt} (V') &= -\lambda (F_k + F_V V') \\
                  &= -\lambda (- V' f'(k) + V' \rho),
\end{align*}
for some $\lambda \geq 0$.
Since $u'(c^*) = V'$,
\begin{align*}
\frac{d}{dt} \log (u'(c^*)) &= -\lambda (F_k + F_V V') \\
                  &= \lambda ( f'(k) - \rho),
\end{align*}
which is an Euler equation.
(Incidentally, the ODE $F(k, V, V')$ does not seem amenable to guess-and-verify, even when $u(c) = \log c$.)


Is zero inflation desirable? Is zero inflation really desirable?

To be more precise: Does inflation in real life have benefits that in some situations outweigh its social cost? E.g.: it works as a disincentive against holding money. The treasury derives income from printing new bills, which usually inflates prices.

The optimal level of inflation is very debated with unclear answers. There are many reasons, and a great answer would be very long. It should also distinguish between expected inflation and surprises.

I'm not going to do any of this, but giving you three reasons for a desirable positive level of inflation. This list is of course incomplete, also there are many reasons against too high inflation.

Downward-rigidity of wages

It is really unclear to economists why it is happening, but nominal wages seem to be downward rigid. It appears to be a behavioral thing (and might not be true at all, see Barro (1977)), but it appears that in crises, once presented with the choice of more firings or reduction of wages, most firms/workers decide to not cut wages but rather respond to the slump with increased separations.

To the extent that we believe this lack of reduction in wages is suboptimal, the central bank can enforce a reduction in real wages through increased inflation rates, and thus preventing separations to some extent.

Redistribution

This argument is based on Keynesian theory. Keynes claimed that the marginal propensity to consume out of income is smaller for rich households (and indeed, we do find this in the data  to some extent). Unexpected inflation is similar to redistribution from debtors to borrowers, as long as debt contracts are not indexed to inflation.

To the extent that poor people consume more out of this unexpected wealth shock and than rich people decrease their consumption, this redistribution will lead to an increase in aggregate consumption.

Room to cutback interest rates

This argument was most prominently brought forward by Krugman around 2008. In recessions, you want to be able to decrease nominal interest rates in order to punish households and firms for "holding cash" and incentivize them to spend it instead. If you start with low (say, 2-3%) nominal interest rates during normal times, you don't have a lot of space to cut back nominal interest rates during the crises.

If, instead, you would have higher nominal interest rates (Krugman argued for around 8%) and relatedly higher inflation during normal times, you could easier cut back nominal interest rates during busts and stimulate the economy.


What's the role for official intervention in the Balance of Payments? If the Balance of Payments (BP) is defined as
$$BP=\text{Current Account}+\text{Capital Account}$$
where:
$$\text{Current Account}=X-M+\text{Net Income Interest}$$
and
$$\text{Capital Account}=\text{F}-\Delta \text{R}$$

Assume current account is positive, and $\text{Net Income Interest}=0$ for simplifying. Then there are 2 remaining ways for the balance of payments to be theoretically equal to zero:

1)$\text{F}<0$ and $\Delta R=0$ i.e. by home country acquiring foreign assets, or

2)  $\text{F}=0$ and $\Delta R>0$ i.e. by the domestic Central Bank (CB) to increase their foreign exchange rate reserves.

Why would there be a need for official intervention by having the domestic central bank (CB) to increase its foreign reserves, if there was no buying of foreign assets ($\text{F}=0$), of ?

Any help would be appreciated.

The Balance of Payments is supposed to be a form of double-entry bookkeeping, in that every transaction has two opposite effects, often in different accounts.  Hence the word balance. In fact it might be considered quadruple-entry bookkeeping, as there are another two effects on the foreign side too.

For a country's Current Account to be in surplus there must be some net income, for example from the sales of exported goods and services exceeding the purchase of imported goods and services.

The surplus could be represented by a debt owed by the foreign importer to the domestic exporter. If the exporter uses this to invest in the foreign country then this counts in the Capital Account (or the Financial Account as the IMF calls it) which balances the Current Account.  But even if it simply remains as debt or as the exporter's balance in a foreign bank, this still counts as other investment and balances in the same way.  If the foreign importer pays in domestic currency then that reduces the outstanding liabilities to foreigners, again with the same net effect on the Financial Account.

If somebody actually turns a debt into money, then it amounts to an investment in the liabilities of the foreign money-issuing authority (typically its Central Bank).  Bringing the foreign money home and selling it to somebody else domestically for domestic currency makes no difference to the impact on the Current or Financial Accounts, unless you sell foreign currency to the domestic Central Bank in which case it turns into a change in the Central Bank's Reserves.


If protectionist policies hinder economic growth, why do we need border taxes at all? I understand that free movement of labour and goods are a net positive for economic growth, and protectionist policies, which would make exports and imports harder would not be.
But if this is the case, why do we need any border tax?
Like in the European Economic Area, where there is a free flow of labour and goods, there is no tax to bring something form one country to another, but if it comes from e.g. Russian there is. Would it make sense to get rid of that tax too, if every single country in the world would do it too?

From pure trade theory perspective there are cases where tariffs can improve welfare of a country provided it is big economy (economy that can affect international prices). And there can be also other arguments made in favor of some protectionism such as infant industry argument (protecting new industry until it is competitive), national security (you don't want to sell arms to your enemy - it would be foolish if US would let its firm sell intercontinental ballistic missiles to Iran), and tariffs can be useful for countries with week central governments which might have easier time to collect taxes on borders (see Krugman et at International Trade: Theory and Policy for overview of these arguments and others).
However, the above being said most economists believe that there should be free trade. Krugman (Nobel Prize winner for his work area of international economics) put this best when he written in his 1987 paper "Is Free Trade Passe?":

If there were an Economist's Creed, it would surely contain the affirmations "I understand the Principle of Comparative Advantage" and "I advocate Free Trade."

The reason why this is so is that even if tariffs can be sometimes improve welfare of a country, these results are contingent on other countries not retaliating (so you don't have trade war), or would be very hard to implement politically (often infant industry argument is abused by firms that do not need it but have political clout) and so on.
Hence generally economists prefer free trade. This consensus still holds, the paper above is from 1987 but more recent poll amongst the top policy economists show overwhelming consensus for free trade. This is why organizations such as WTO are set up with explicit goal of reducing tariffs and other trade barriers across world.

This being said, most economists would still argue that there are exceptional situations. For example, as mentioned above you don't want to sell arms (or crucial component of military equipment) directly to your enemy (although this can be still misused - Trump administration used this argument when it implemented  Huawei chip ban and this was criticized by some economists). WTO allows for increases in tariffs for countries that break trade treaties or raise their own barriers unliterary (as sort of deterrent).  Recently, some good arguments have been made in favor of environmental tariffs which would be applied to countries that do not have their own environmental policy.
In a specific case of Russia and EU there can be arguments for tariffs on the above mentioned environmental grounds or as a response to aggressive international policy that Russia is pursuing (landgrabs in Ukraine, threatening sovereignty of some Baltic EU members). However, this being said in principle if there would be free trade area between EU and Russia (with exception of dirty goods or some common environmental policy) it would be to the net benefit of both parties in long-run (if we ignore the political issues).


How Does GDP Differ From NNW as an economic measure? When people talk about the size of an economy, they often refer to the GDP or changes to it, but seldom refer to the Net National Wealth. Obviously there are some superficial differences:

GDP is Gross not Net
GDP is Domestic not National (Within borders vs. owned by citizens)
GDP is $\Delta$Capital whereas NNW is theoretically all the capital stock owned by a country's citizens.

It appears the US switched from publishing GNP to GDP relatively recently (~1970), which seems like a less direct measure of its citizens economic welfare, but even that seems like an already abstracted measure from NNW.

So again, what are the differences between the two measures as economic indices that makes one more desirable (or at least discussed) than the other.

I cannot find any reference to a specific "Net National Wealth". But I assume that you refer to the National Wealth, in general (also called Net National Worth).

The GDP has been explained in another question, so I will not come back to it.

The National Wealth is a measure of the total wealth of a country: assets, liability, etc.

To my understanding, the main difference resides on the fact that the NW is a rather static measure, whereas the GDP focuses on the dynamic part. Some examples

the value of a house will appear in NW, but not in the GDP;
if you buy a product from your neighbour, the NW will stay constant, but the GDP will increase.

If a country has a high GDP, its citizens/companies should accumulate wealth, and thus the NW should increase. But as the GDP also contains movement within the country those are not reflected in the NW. A diamond is in NW, the operations to cut it is in the GDP.


Why am I getting 3 different answers for Point Elasticity of Demand? First year Econ. Question:

Demand function: $P=50-4Q$ (Price in cents/kilo)

Supply function: $P=25+Q$

Equilibrium Price and Quantity: 30 cents, 5 kilos.

Price change: 40 cents, 2.5 kilos.

I've tried 3 different methods of calculating point elasticity:

$\frac{\% \Delta Q}{\% \Delta P}$ =  $\frac{-50\%}{133\%}= -.375$;
$\frac{\frac{Q_{1}-Q_0}{Q_0}}{\frac{P_1-P_0}{P_0}} = \frac{\frac{2.5-5}{5}}{\frac{40-30}{30}}=-1.5 $
$\frac{1}{slope}\frac{P}{Q}=\frac{-1}{4}\frac{40}{2.5} = -4$

The right answer is -4, but why the variability?

In your second option, you computed the elasticity at the original point, and in your third option, at the end point. In your first point, you did the same as in the second one, but with a mistake: you wrote $133%$ instead of $33%$.


What does Marx mean by "the division between capital and labor"? I am reading an eight page piece called "Labor Alienation" by Karl Marx for my Literature class. Here's a link to it if you would like to read it:

http://www.yavanika.org/classes/reader/marx1.pdf

In it there is a line that says:

  Political economy does not disclose the
  reason for the division between capital and labor, between capital and land.

My background in economics is quite basic, so I apologize for this very basic question. But whatever help I can get will be greatly appreciated!

Two main changes happened when capitalism came into existence these being the ability to buy and sell property among the peasantry.

Under the feudal system each household was endowed land which was used to produce food/ agricultural products for the household and the lord which provided the land.

When feudalism was abolished the possibilities of production outside the land allotted to the former peasantry was an option. However the nature of the work changed in the sense that workers were still productive but did not own their means of production in the same way that they did under the feudal system. they worked with the machines and materials provided by their employer but did not take the product back, but rather settled for a working wage which is systematically less than his own work. This is exactly the split between land and labor Marx is talking about- the inability of the worker to have access to the true product of labor (or value of it).

The land provided by the lord was a place where the peasants lived so they were always one with their means of production and benefited from it. In the new capitalist system this was no longer possible.


Finding demand function given a utility min(x,y) function I am confused about a particular point regarding finding a demand function. All the problems in this practice set I am doing have involved applying the method of Lagrangian multipliers. But I am uncertain if it applies here for this problem.

Problem Setup

Consider a consumer with utility function $u(x,y) = \min\lbrace x,y\rbrace$. Suppose we are given wealth $w$ and prices $p_x = 1, p_y = \frac{1}{2}$.

My Work

Not much to do yet. All I did was set up a budget constraint $w = xp_x +  yp_y = x + \frac{1}{2} y$.

My Confusion

I was all set to setup a Lagrangian multiplier equation when suddenly I realized that my utility function is a $\min$ function. At first, I thought this function wasn't differentiable. Now, I am thinking it is not differentiable but it is partially differentiable. I am still unsure.

My Guess

I suspect yes $\min$ is partially differentiable based on this thread

https://math.stackexchange.com/questions/150960/derivative-of-the-fx-y-minx-y

But I suspect my answer will need a piecewise component or something.

My Question

Are Lagrangian multipliers applicable here? If so, how do I define the Lagrangian in piecewise terms as I think I will need to do?  If it is not differentiable, how does one derive a demand function given a $\min$ or a $\max$ function?

No, you should not use Lagrange multipliers here, but sound thinking. Suppose $x\neq y$, say for concreteness $x<y$. Let $\epsilon=y-x$. Then $\min\{x,y\}=x=\min\{x,x\}=\min\{x,y-\epsilon\}.$
So the consumer could reduce her consumption of good 2, without being worse off. On the other hand for all $\delta>0$, we would have $\min\{x+\delta,y-\epsilon/2\}>x=\min\{x,y\}$, so the consumer could be better of by reducing the consumption of the second good and spending the freed money on the first good. In an optimum, a consumer cannot improve so optimality requires $x=y$. It is also clear that consumers improve along the $x=y$ 45° ray. So you can simply use $x=y$ as an optimality condition to be substituted into your budget constraint and bypass Lagrange multipliers.


Perfect Substitution with a Continuum of goods? Imagine I have a continuum of different goods indexed by $\omega \in [0,1]$. I have a household which consumes a quantity $C(\omega)$ of good $\omega$, and pays a price $P(\omega)$. The household has a budget equal to 1, and has a linear utility function.

$$U(C) = \int_{0}^1 C(\omega) d\omega $$

The consumer chooses her consumption bundle by solving the problem

$$\max_{C} U(C) $$
$$ \text{Subject to}:  \int_{0}^1 P(\omega) C(\omega) d\omega \leq 1.$$

My question is with regards to the fact that this problem seems ill-defined. If I were to go ahead solve the optimization problem, I would have the consumer choose

$$C(\omega) = \begin{cases} \frac{1}{P(\omega)} \text{ if } P(\omega) =  \min_{\omega'} P(\omega') \\ 0 \text{ otherwise.} \end{cases}$$

However, this means that  $U(C) = \int_{0}^1 C(\omega) d\omega = 0$, and $\int_{0}^1 P(\omega)C(\omega) = 0$ since $C(\omega) = 0$ almost everywhere.

Is there a mathematically more robust way of defining this problem? One approach I've seen is using Dirac delta functions, but I'm not sure that's right. That is, defining

$$U(C) = \int_{0}^1 C(\omega) \delta_{\omega} d\omega$$
where $\delta(\omega)$ is a ``function'' which is equal to $+\infty$ at $\omega$ and $0$ everywhere else.

Seems like the problem specified as specified right now should have as solution $C(\omega)=\frac{\delta(\omega)}{P(\omega)}$ if $P(\omega)=\min_{\omega'} P(\omega')$ and zero otherwise. The solution proposed cannot be optimal because the budget constraint is not depleted. If we let $\omega^*$ be the index of the cheapest good, the indirect utility would be then $U^*(P(w))=1/P(\omega^*)$.


What was happening in Madagascar during 1971-1996? In the course of my research I have been examining these days macroeconomic time series for per capital household consumption, using the databank of the WorldBank. The unit of measure was constant 2005 USD.

I got all shorts: smooth upward trends indicating a steadily increasing material well being on average; series with wild swings indicating socioeconomic turmoil, civil wars, coup d'eta's etc; series with an obvious structural shock in the level which then continued to grow at the same rate (for example Finland experienced such a shock when the Soviet Union collapsed); series suspiciously smoothly increasing, too smoothly to be believable and so suspect of being fake...

Unique among them, was Madagascar: its per capita consumption exhibited a steady downward trend for the period 1971-1996, although after that period the trend seems to (gradually) flatten out. Here is the plot:

Again the unique feature here is the almost constant downward trend. There were other countries that saw their consumption level collapsing in this period, but not in such a steady gradual manner, over a quarter of a century.

Does anybody have a clue as to what was happening in the period in this country? Madagascar was colonized by France in 1896 and obtained full independence in 1960. But of course there are many countries that were colonies and de-colonized around that period; none exhibited such behavior.

I think I have found the explenation for this. If you look at the wikipdedia page of the economy of Madagascar, it states the following:

The standard of living of the Malagasy population has been declining dramatically over the past 25 years. The country has gone from being a net exporter of agricultural products in the 1960s to a net importer since 1971. Inappropriate traditional agricultural methods cause soil to erode and soil quality to decline, and the basis of survival for Madagascar’s people is under serious threat.

The source of this claim is Deutsche Gesellschaft für Internationale Zusammenarbeit.

Now, if you look at the world bank figures on agriculture in Madagascar, you can see this declining trend. For example the Agriculture value added per worker (constant 2005 US$) declines between 1980 and 2014 from 250.1 to 175.8.

Since Madagascar is a country which relies heavily on agriculture (about a quarter of its GDP stems from agriculture), this declining trend influences the whole of this economy.


Multivariate cointegration methods in Econometrics I've studied a number of cointegration methods in the past few months. It seems, Johansen's multivariate VAR approach to cointegration is the most commonly used multivariate cointegration technique. Most packages appear to have in-built functions to run "Johansen cointegration analysis".

I was wondering if there are other multivariate cointegration methods in the literature though less known and are not available in let's say Eviews, Stata, RATs etc. but can be programmed.

I think the main alternative to the Johansen (statistical) approach is the methodology propounded by Pesaran and Shin in their so-called long-run structural modelling (economic) approach. The main formal reference is Pesaran and Shin (2002). The methodology is also presented to a wider audience in Garratt et al. (2012).

Although referred to as the long-run structural modelling approach, you will also read about the type of models associated with the methodology, which are called VARX* models. The distinguishing feature is that the cointegrated models are estimated using reduced-rank regression and (over-)identifying restrictions are derived from economic a priori then tested. The sequencing is slightly different to Johansen since economic theory is given priority.

In the same school of thought is the cointegrated VAR approach associated with Juselius (2006). Again, Juselius propounds an economic approach to cointegration as opposed to the statistical approach by Johansen (and others like Phillips).

Pesaran's approach has been programmed and is available in the Microfit software.

Juselius' approach has also been programmed and is available in the RATS (CATS) software.

To my understanding, you can find MATLAB code in the GVAR toolbox, and this should give an indication of what's required for the Pesaran approach. In terms of programming, I haven't seen much difference when compared to Johansen (although I just quickly investigated this). Note that the GVAR approach can be thought of as an extension to the VARX approach (the difference lies in stacking individual country models, but estimation of the individual models is the same).

References

Pesaran, M and Shin, Yongcheol, (2002), LONG-RUN STRUCTURAL MODELLING, Econometric Reviews, 21, issue 1, p. 49-87.

Garratt, Anthony, Lee, Kevin, Pesaran, M and Shin, Yongcheol, (2012), Global and National Macroeconometric Modelling: A Long-Run Structural Approach, Oxford University Press.

Juselius, Katarina (2006) The Cointegrated VAR Model: Methodology and Applications (Advanced Texts in Econometrics).

GVAR Toolbox


Analysis of free goods The normal supply and demand models take the supply and demand of a particular good and show that the equilibrium price is where the two curves intersect.  At this point, all the people below the equilibrium point value the good as worth less than the price; on the other hand, all the people above the equilibrium point value the good as worth more than the price; people at the equilibrium point value the good at the price value.
In my mind, following this logic, everyone should want to "purchase" a free good.  But that is not necessarily the case.  For example, in an undergraduate program, the university will give away t-shirts, pens, etc. to the students for free.  However, not all of the students will want these items.  However, since the price is free, it seems that everyone should want them.
How do we reconcile both of these observations?

You can imagine a demand curve continuing through the x-axis as monetary price becomes negative. A negative price can be interpreted as payment to pick up shirts, a kind of subsidy.* That might tempt those who are still unwilling at price of zero because there's still a opportunity cost of picking up the shirt, or because they only wear collared shirts, or because disposal is not free (the unwanted shirt would need to recycled).

In the simple model, the demand curve that represents willingness to pay is holding these other factors fixed. In a more complicated model, the price drop might induce congestion at the student union, so the demand curve would also shift down or up, depending on your desire to spend time in a crowd of college students.

*Obviously, you have to impose some sort of limit per student to keep quantity demanded from shooting off to infinity.


Understanding Vernon Smith's 1962 "An Experimental Study of Competitive Market Behavior" Vernon Smith's 1962 paper "An Experimental Study of Competitive Market Behavior" simulates a market using students (who have piece of paper giving their reservation prices). In one of the experiments, the demand/supply curves look like the graph on the left, with the corresponding market activity on the right:

Since there seems to be nothing special about using students in the experiment (i.e. as far as I can tell, they are just performing simple computations), I decided to code up the experiment in Python. My script is here.
When I run the script (which has the same demand/supply curves as above), the market activity typically looks like this:

The market price does indeed hover around $2.00, but my graph seems noticeably more jagged than the graph in the original paper. I'm trying to figure out what's going on, in particular whether:

I'm understanding the paper incorrectly somehow, or
My Python script has some error in it, or
There's some kind of error or unwritten assumption in the paper.

I suspect that the reason for that result is the way how you modeled the situation is by having the prices to be offered at random from a sample of numbers in uniform fashion (unless I am misreading the code) but this is unrealistic.
As mentioned in the paper Smith's 1962:

each sequence of experiments was conducted over sequences five to ten minutes long

So the students had only 5-10 minutes to actually conclude the trades. Students knew their own valuation of the item and they wanted to get it.
In such situation it is not in students interest to just choose random price from the sample of prices. Rather students will be choosing from some distribution that will have more mass closer to the their own valuation.
My prediction is that if instead of random sampling of prices from the array you will let the program generate numbers from some distribution that has more mass closer to the right price you will see less volatility in the data.
For example, instead of using random.choice(price_deltas) you could use random.choice(price_deltas, p=weights) that allows you to assign weights to numbers so some are more likely than others.

EDIT:
You could use the weight approach above but I also discovered much simpler one. If you are just fine assuming prices can be continuous (which from point of economic theory is fine 0.01c is really just practical limit in real life), then what you can do is skip the creation of deltas at the start:
import numpy as np
import matplotlib.pyplot as plt
buyers = [3.25 - 0.25*n for n in range(0, 11)]
sellers = [3.25 - 0.25*n for n in range(0, 11)]

Next what you can do is to replace random.choice() with some distribution that has most mass at zero - like the exponential distribution. In python this can be done using random.exponential(), the code would thus change only minimally, and will generate deltas that are closer to the actual valuation of the traders:
trading_prices = []
trading_counts = []

for _ in range(5):
    buyers_still_in_market = list(range(len(buyers)))
    sellers_still_in_market = list(range(len(sellers)))

    transactions = []

    for _ in range(1000):
        # First, a random buyer offers to buy at some price below their number on
        # their card
        b = np.random.choice(buyers_still_in_market)
        delta = np.random.exponential(400)
        buyer_price = max(0, buyers[b] - abs(delta))
        s = np.random.choice(sellers_still_in_market)
        if buyer_price >= sellers[s]:
            transactions.append((b, buyers[b], s, sellers[s], buyer_price))
            buyers_still_in_market.remove(b)
            sellers_still_in_market.remove(s)
            trading_prices.append(buyer_price)

        # Then, a random seller offers to sell at some price above their number on
        # their card. #0.65 - for log normal
        s = np.random.choice(sellers_still_in_market)
        delta = np.random.exponential(400)
        seller_price = sellers[s] + abs(delta)
        b = np.random.choice(buyers_still_in_market)
        if seller_price <= buyers[b]:
            transactions.append((b, buyers[b], s, sellers[s], seller_price))
            buyers_still_in_market.remove(b)
            sellers_still_in_market.remove(s)
            trading_prices.append(seller_price)

    trading_counts.append(len(trading_prices))

    print(len(transactions), "transactions")

plt.plot(list(range(len(trading_prices))), trading_prices)
plt.ylim(0, 4.00)
for x in trading_counts:
    plt.axvline(x=x-1, ymin=0, ymax=4.0, color='red')
    print("line drawn at x =", x)
plt.show()

print(trading_prices)

This new code produces results that are much less 'jagged'. Here are some examples, of behavior of price with the exponential distribution (possibly different types of distribution would yield better result but its beyond scope of SE answer to examine that). For example, here are 8 graphs generated when the prices depend on exponential distribution:


If an economy has capital that is less than the golden rule level of capital, can we reach the golden rule without increasing the savings rate? This question is in context of this problem I was trying to solve:

The answer given is (a). But I think the answer is (C). My arguments are as follows:

First, the golden rule level of capital corresponds to one and only one savings rate, $ s^* $.
Thus, to reach $ s^* $, we need to increase the savings rate (since the economy starts off below the golden rule of capital accumulation steady state).
Increasing savings reduces consumption initially.
But increased savings stimulate investment, which increases production, which ultimately increases income and thus **increases consumption in the long run.

Is my answer correct, speaking strictly in terms of academic Solow Model theory and with the information given in the question? If wrong, why?

Yes the answer should be C. I have attached an image showing the variation with time of the variables $y$(per capita output), $c$(per capita consumption) and $i$(per capita investment). I am assuming at $t=t_0$ the savings rate is increased.

The consumption per capita initially will fall because the savings rate has increased. Eventually it must go above the initial per capita consumption because of the Golden Rule condition.

Cheers!


Difference between $x$ and $X$ Sorry for my bad english, i'll try to explain my difficulty.

1° Question

In the last exam an exercise asked to derivate OLS estimators and calculate their value with the data provided (ex.: $\sum x_iy_i=...$, $\sum Y_i=...$). Now, I know how to derivate them ($\hat{\beta _{0}}=\bar{Y} -\hat{\beta _{1}}\bar{Y}$ and $\hat{\beta _{1}}=\frac{\sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y})}{\sum_{i=1}^{n}(X_i-\bar{X})^2}$) but i have difficulties calculating them because data can be "big" ($X$) or "little" ($x$). My prof. tried to explain me that $x=X-\bar{X}$ but my difficulties remain. For example: assuming that $\sum x_iy_i=...$, $\sum Y_i=...$, $\sum X_{i}^{2}=...$ and $\sum X_i=...$: all data are uncertain except the last. I know that $\hat{\beta _{1}}$ can be written like
$=\frac{\sum_{i=1}^{n}X_iY_i- n\bar{X}\bar{Y}}{\sum_{i=1}^{n}X_{i}^{2}-n\bar{X}^2}$: so, how can i transform $x_iy_i$ in $X_iY_i$ to use it in the formula?

2° Question

In light of the problems that i have for the 1° question, i can't write a good form so i ask you if formulas below are corrected:

mean for observed variables: $\bar{x}=\frac{1}{n}\sum x_i$
variance for observed variables: ${s_{X}}^{2}=\frac{1}{n}\sum (x_i-\bar{x})^2=\frac{1}{n}\sum ({x_{i}}^{2}-2x_i\bar{x}+\bar{x}^2)=\bar{x^2}-\bar{x}^2=\frac{1}{n}\sum {x_{i}}^{2}-\bar{x}^2$
standard deviation for observed variables: ${s_{X}}^{2}=\sqrt{\frac{1}{n}\sum (x_i-\bar{x})^2}$
covariance for observed variables: ${s_{XY}}=\frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})=\bar{xy}-\bar{x}\bar{y}=\frac{1}{n}\sum x_iy_i-(\frac{1}{n}\sum x_i)(\frac{1}{n}\sum y_i)$
correlation for observed variables: ${r_{XY}}=\frac{s_{XY}}{s_{X}s_{Y}}=\frac{\frac{1}{n}\sum x_iy_i-(\frac{1}{n}\sum x_i)(\frac{1}{n}\sum y_i)}{\sqrt{\frac{1}{n}\sum (x_i-\bar{x})^2\cdot\frac{1}{n}\sum (y_i-\bar{y})^2}}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2\cdot\sum (y_i-\bar{y})^2}}=\frac{\sum x_iy_i-n\bar{x}\bar{y}}{({\sum x_{i}}^{2}-n\bar{x}^2)({\sum y_{i}}^{2}-n\bar{y}^2)}$
sample mean: $\bar{X}=\frac{1}{n}\sum X_i$
sample variance: ${S_{e}}^{2}=\frac{1}{n}\sum (X_i-\bar{X})^2$
standard error: $SE=\sqrt{\frac{1}{n}\sum (X_i-\bar{X})^2}$
corrected estimator for sample variance: $S^2=\frac{1}{n-1}\sum (X_i-\bar{X})^2$ with $S^2=\frac{n}{n-1}{S_{e}}^{2}$
$R^2$: $R^2=(r_{XY})^2$ or $R^2=\frac{ESS}{TSS}=\frac{\sum_{i=1}^{n}(\hat{y_i}-\bar{y})^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}\Rightarrow ESS=R^2\cdot TSS=(r_{XY})^2\cdot (\sum {y_{i}}^{2}-n\bar{y}^2)\Rightarrow RSS=TSS-ESS$
estimate of the variance of residuals: $SER=\sqrt{\frac{1}{n-2}\sum (e_i-\bar{e})^2}=\frac{RSS}{n-2}$

Formulas can be checked against infinite online resources, not here.

The most widespread use of $x$ and $X$ to distinguish something, is when it is needed to emphasize what is treated in theoretical derivations as a realized value (a fixed number, $x$), and what as a random variable, $X$. When using matrix algebra though bold uppercase denotes a matrix (but in panel data where everything is a matrix, we see often simple uppercase to denote a matrix).

Sometimes, authors don't bother, since it should be obvious from context.

Perhaps the most usual confounding is to use lowercase $x$ for either a realized value or a random variable, but use uppercase letters for random variables.

As for calculation, it is always done using data, so there is nothing to "convert" here. If you have the realized data on $X$ and $Y$, then plug them in the formula.

Any other contrasting use of the two is of course not "forbidden", but then one should read carefully how each author defines their use. Nevertheless, to use simply $x$ to denote "a variable centered on its sample mean" $x=X-\bar X$, is rather unusual and confusing I would say, and in any case it does not warrant a change from lowercase to uppercase. We usually write things like

$$x^* \equiv x-\bar x$$

or some other symbol instead of a star.


Amazon vs Netflix, closely resembles which type of competition? So we all know Amazon is following the path of Netflix regarding tv shows and movies. Both firms are increasing production of film, and now have their own productions.

If we were to categorize this type of competition, would it more closely resemble Hotelling's Model or Cournot?

It certainly isn't a Cournot scenario. That would require homogeneous products. In fact there's a ton of differentiation between the two products (Amazon prime and Netflix). They have different TV shows, different movies, different documentaries.

I would argue that it is this differentiation that allows them to charge different prices.

Hotelling's law is much more applicable. However I would argue that they do not maintain minimum differentiation. Yes they may share similarities in their web presence, and they may have adopted the strategy of producing content in house; but they also are so different that it is hard to say they are copying each other on almost everything.

Amazon prime offers reduced price shipping, music streaming, and reduced prices on some items on Amazon. Netflix has none of those things.

That said, Netflix includes hard copies of DVDs as rentals included in their subscription, whereas you would have to buy the DVDs on Amazon. So even though some of their business is similar or matching, they mostly represent differentiated products.


References to learn continuous-time dynamic programming Does anyone know of good references to learn continuous-time dynamic programming? The references don't have to be books. They could be links to online resources as well. Links to clear, concise discussions of even just the basics would be helpful.

For continuous-time stochastic dynamic programming, the small, nontechnical Art of Smooth Pasting by Dixit is a wonderful option. It does a very effective job of conveying the basic intuition.

Stokey's more recent The Economics of Inaction is also decent, but for a practical-minded person it probably underperforms Dixit - its much greater length and somewhat heavier notation do not yield commensurate rewards.

If the underlying stochastic processes are not Itō diffusions, then I'm not sure what the best reference is. The most common such case I've seen (and that I use myself) is the case of discretely many exogenous states, where if we are currently in state $s$ there is some constant hazard rate $\lambda_{s,s'}$ of a switch to state $s'$. Fortunately, this is a pretty simple case in practice: one can just modify the HJB equation to account for the flow probability of switching from $V(\cdot,s)$ to $V(\cdot,s')$. (You can see this, for instance, in equations (1)-(5) in this Acemoglu and Akcigit paper. Conceptually it's no different from setting up the HJB equation when we have an Itō diffusion as the driving process, except that it's simpler because we just get a system of linear equations and we don't need to think about Itō's lemma, etc.)

Of course, maybe there is a good textbook reference for this too - but unlike the potentially much more complicated cases involving stochastic calculus, this is straightforward enough that a text has never seemed necessary to me.


Contour set in Mas-Collel, Winston and Green Mas-Colell, Winston and Green's Microeconomic Theory uses the notion of contour set, as shown hereunder:

In the context of Microeconomic Theory, what would be the definition of a contour set?

The upper contour set is $ UC(x)=\{ y|y \succsim x \} $, i.e., all the goods baskets that are at least as desirable as $x$. Similarly, the lower contour set is $LC(x)=\{ y|x \succsim y \} $, i.e., all the goods baskets that are at most as desirable as $x$.


Straffin, P. D. (1993): Game Theory and Strategy. How is the security level on p. 103 calculated? Could somebody explain how Straffin calculated the security levels for Rose and Colin on p. 103? As far as I understood, Colin's security level is 6 because it is the minimum of the strategy Colin A because Colin A dominates Colin B. If both, Colin A and Colin B were considered, Rose should follow a mixed strategy of 8/9 for Rose A and 1/9 for Rose B, which leads to a security level of 20/9, but only if Colin A was considered. I don't understand, how Straffin came to the security level of 10/3 for Rose and why the strategy BA should be played at least 1/3 of the time.

Here are the two pages that describe the problem (I apologize for the bad resolution):

Suppose Colin plays A with probability c and B with probability 1-c, while Rose plays A with probability r and B with probability 1-r. Then Rose's payoff is

$2rc+4(1-r)c+10r(1-c)+0(1-r)(1-c)$

Rose's security level is calculated by taking the max-mini payoff: that is, for each value of r, find the minimum payoff over all c, then find the maximum such value.

So, first, we can simplify the above expression:

$2rc+4c-4rc+10r-10rc = 4c+10r-12rc$

As a function of c, this is a line with intercept $10r$ and slope $4-12r$.

Case I: $r = \frac13$, this line is a constant $10r$. Plug $r =\frac13$ in and you get $\frac{10}3$

Case II: $r > \frac13$, then it's negatively sloped, and the minimum occurs at $c = 1$, with a payoff of $4-2r$. Since $r > \frac13$, we have that the payoff is less than $4-2/3 = \frac{10}3$.

Case II: $r < \frac13$, then it's positively sloped, and the minimum occurs at $c = 0$, with a payoff of $10r$. Since $r < \frac13$, the payoff is less than $\frac{10}3$.

In other words, if Rose plays A $\frac13$ of the time, then she gets an average of $\frac{10}3$ regardless of what Colin does. If she plays A less than $\frac13$ of the time, then if Colin plays B, then she gets 10 less than $\frac13$ of the time and 0 the rest of the time, getting her less than  $\frac{10}3$. If she plays A more than $\frac13$ of the time, then if Colin plays A, she gets 2 more than $\frac13$ of the time and 4 the rest of the time, getting her less than $\frac{10}3$.

So Rose can guarantee herself $\frac{10}3$ by playing A $\frac13$ of the time, and every other strategy has a possibility of giving her less than that. Thus, $\frac{10}3$ is her security level.

The reason BA should be played at least $\frac13$ of the time is that if BA is played  $\frac13$ of the time  and AB is played  $\frac23$ of the time, then Colin's payoff is  $8*\frac13+ 5*\frac23 =  \frac83 +  \frac{10}3 = \frac{18}3=6$, which is his security level. If BA is played less than  $\frac13$ of the time, then his payoff goes below his security level, so he would be better off not cooperating.


Is it possible to transfer information through strategic games? Let's say that there are 2 players Player A and Player B, engaged in a set of repeated strategic games. Is it then possible to transfer information from one player, say Player A, to the other player, Player B, simply by having Player B observe the strategies that Player A uses over a given period of time, in response to Player B's strategies?

For example, Player B could form a simple hypothesis about player A's beliefs  and/or preferences, and test it out by using different strategies. The hypothesis can then turn out to be true or false (over a period of time, with some probability) if Player A's strategies in response to Player B's take on a certain character.

I would like to learn more about these kinds of games, if they've been studied.

What you are implying is thoroughly studied by what is called algorithmic game theory. There you can find the concept of a learning algorithm. In game theoretical scenarios where the mathematical equilibrium is difficult to evaluate, those algorithms are used to get an approximation.

What they do (in a simplified way)
1) Player A starts with an array of available strategies, each to be played with equal possibility.
2) In every iteration he randomly chooses one strategy and plays against his opponent.
3) The result comes as a feedback and according to an update function/algorithm, the array of strategies is re-evaluated/recalculated.
4) This is repeated many times, hopefully to converge to either one strategy which will be a nash equilibrium, or to a mixed-strategy profile.

It is a very interesting topic, connected a lot with machine learning and AI.
Here are some links to get an idea:

1) Game Theory & Learning for wireless networks

2) On Learning Algorithms for Nash Equilibria


Efficiency in a market that is both a monopoly and a monopsony We know that market power in general leads inefficient outcomes (in terms of Pareto efficiency). This makes instinctive appeal as, for example, a powerful seller who has a lot of market power can benefit by restricting supply, single-handedly raising prices.
But I was wondering, what if there are not many consumers either and thus, there is also a lot of monopsony power?
In the extreme, how is equilibrium determined in a market with just one seller and one buyer? What are its efficiency properties.
I have a feeling it is just bargaining, and maybe related solutions like Nash bargaining solution might help. Am I right? I am not sure on the efficiency implications though.

Efficiency in general means that the item being traded should be given to the party that values it the most.
In a competitive market, this means trade should continue as long as a consumer's value of a good, as captured by the demand curve, is greater than a seller's value of that good, captured by the supply curve.
In a monopoly, seller's value is captured by its marginal cost curve. But since price is greater than marginal cost in equilibrium, the units for which a consumer's value is higher that MC but lower than equilibrium price don't get traded. Hence there is inefficiency: those units of the good are not in the hands of the party (i.e. the consumers) that values them the most. The story for monopsony is similar.
In a bilateral trade setting with one seller and one buyer, we need to discuss two cases. In the first case, the traders' values are common knowledge. In this case efficiency can be attained, though price depends on the relative bargaining power of the two parties. In the second case, suppose neither trader knows the other party's value, but they do know that gains from trade are possible but not certain. Then the Myerson-Satterthwaite theorem tells us that no bilateral trading mechanism that is individually rational and incentive compatible can guarantee efficiency in the ex post sense; that is, the party with a lower value for the good being traded may end up having the good.


When Does a Shift to the Right Happen in the Demand and Supply Curves Simultaneously? If, for example, Health Canada issues a public statement that states eating pizza is good for your overall physical health, would this result in both a shift in the demand curve for pizza to the right, and a shift in the supply curve of pizza to the right?

If not, could you give more information on when this would actually happen?

To answer your first question: No, it shifts out only the demand curve but not the supply curve. However, the quantity supplied increases.

To understand why the advertisement does only shift out the demand curve but not the supply, you have to understand two important concepts:

(1) Shift of the demand curve = Increase of demand;
(2) Movement along the demand curve = Increase of quantity demanded.

Similarly with supply:

(1) Shift of supply curve = Increase of supply;
(2) Movement along the supply curve = Increase of quantity supplied.

Now what happens, if you advertise for pizza? The demand as well as the quantity supplied increase. To see this, consider the following. More people demand pizza so the demand curve shift out and the demand increases. At the same time, the supply curve does not move. BUT, the outwards shift of the demand curve intersects the supply curve further to the right, and hence, the quantity supplied increases.

Shifts of the demand curve can be caused, for example, when people's income increases or decreases. Demand increases usually increase the quantity supplied.

Shifts of the supply curve can be caused, for example, through the the use of better technology that lowers producers marginal production costs. Supply increase usually increases the quantity demanded.

Increases or decreases of quantity demanded or quantity supplied can be caused through price changes.

A simultaneous increase of demand and supply can be cause if consumers get a higher income and firms get a better technology at the same time.


What are general equilibrium effects? Im reading alot on public finance and have been seeing alot of mention of measuring "general equilibrium effects".

I know what general equilibrium is, but i dont know what general equilibrium effects are.

What are general equilibrium effects?

The expression "general equilibrium" refers to the work by Kenneth Arrow, Gerard Debreu, Leon Walras, and many others. In its strictest sense, it is the indirect impact of a policy/event that is due to the reaction of prices to the policy/event. Sometimes but not always this indirect impact offsets the beneficial impacts of the policy or event.

Sounds abstract? Let me give you an example.

An oil field is discovered in a small town in Texas. As the oil exploration company develops the field, that generates jobs and higher incomes. Looks great, because local businesses will enjoy higher revenues and unemployed workers may find employment.

Yet, there can be some substantial general equilibrium effects that partly offset  the positive consequences of this beneficial event:

The prices of many products will rise. Housing is going to be more expensive as higher-income workers are going to try to purchase houses in the town. Restaurant meals and haircuts will become more expensive.
A positive consequence: there will be fancier restaurants and hair salons in town. The higher incomes will attract more talented chefs and better hairdressers.
There may be more congestion on the road, as more workers means more cars.

Sometimes the general equilibrium effects are such that the beneficial consequences of the event are almost entirely offset (more expensive, more congestion, more pollution). Sometimes the general equilibrium effects are small and can be ignored. If, for instance, there were already a lot of empty homes in this small town, the additional workforce and income won't drive house prices up. Workers will merely buy the existing stock of houses.


Finite's expected utility Let $\succeq$ be a rational preference relation over a non-empty set $X\equiv\mathbb{R}$ and suppose it satisfies:
$$
x\succeq y \implies x+z\succeq y+z,\forall z\in X
$$
Then I want to show that if $x\succeq 0$ then, $\forall\lambda=\frac{m}{2^n}$, we have $\lambda x\succeq 0$.

How can I show that?

Thanks for helping! :D

P.S.: The book I'm using is not really clear on the math hypothesis, especially on what sets everything is defined and so on, so I'm assuming $m,n\in\mathbb{N}$.

Here is a proof by induction on $n$, for any $(n,m) \in \mathbb{N}^2$.

We will need the following preliminary result, which strengthens the assumption on $\succeq$:

Lemma
\begin{equation*}
\forall x,y,z \in \mathbb{R}, x \succ y \Rightarrow x+z \succ y + z
\end{equation*}

Proof
Suppose that $x+z \preceq y+z$. By your assumption on $\succeq$, we can add $-z$ to both sides of this inequality (since $-z \in X=\mathbb{R}$) and obtain
\begin{equation*}
x+z+(-z) \preceq y+z+(-z)
\end{equation*}
i.e. $x \preceq y$. This proves the lemma (by contrapositive).

Now, let us prove the main result.

Basis ($n=0$)

We need to prove that $mx \succeq 0$ for any $m \in \mathbb{N}$. This part can itself be proved by induction on $m$. It is trivial for $m=0$ and $m=1$. If the property holds for $m$, then
\begin{equation*}
(m+1)x = mx +x \succeq mx + 0 \text{ since } x \succeq 0
\end{equation*}
Therefore $(m+1)x \succeq mx$, and since $mx \succeq 0$ by the induction hypothesis, by transitivity we obtain $(m+1)x \succeq 0$: the result holds for $m+1$. This proves the result for $m \in \mathbb{N}, n=0$.

Inductive step

Let us fix $m,n \in \mathbb{N}^2$ and suppose that $\dfrac{m}{2^n}x \succeq 0$. Let us show that $\dfrac{m}{2^{n+1}}x \succeq 0$. To this end, we proceed by contradiction and we assume, otherwise, that $\dfrac{m}{2^{n+1}}x \prec 0$. Our lemma yields
\begin{equation*}
\dfrac{m}{2^{n+1}}x + \dfrac{m}{2^{n+1}}x \prec 0 + \dfrac{m}{2^{n+1}}x
\end{equation*}
and therefore (computing each side of the inequality)
\begin{equation*}
\dfrac{m}{2^{n}}x \prec \dfrac{m}{2^{n+1}}x
\end{equation*}
By assumption $\dfrac{m}{2^{n+1}}x \prec 0$, and we just showed that $\dfrac{m}{2^{n}}x \prec \dfrac{m}{2^{n+1}}x$. By transitivity we obtain $\dfrac{m}{2^{n}}x \prec 0$. This is a contradiction, since the result was supposed to hold for $n$.


Economics of forgetting I'm interested to know economic justifications of forgetting in various fields.

On example is Green and Porter(1984). To sustain a Cartel, Cartel members have incentive to forget deviations in a sufficiently distant past.

Another is Ekmekciy(2011) which shows that a rating system works better if it stops publicize misconducts, once they're old enough.

Literature on a more abstract level is also welcomed, e.g. results like it's more socially desirable in a dynamic games, some agents are restricted to stationary strategies.

Maybe not what you are looking for, but a related concept is regret. Orphanides and Zervos (1995) is a classic paper on rational regret in a health economics concept http://www.jstor.org/discover/10.2307/2138580?uid=3739840&uid=2&uid=4&uid=3739256&sid=21106216442271

There's also plenty of irrational regret papers, mostly just boiling down to hyperbolic discounting models.


Good paper/article on the mechanisms in RBC vs New Keynesian models I have read technical books about the (baseline) RBC and New Keynesian models.

However, these books tend not to explain very intuitively the propagation mechanisms of shocks between the two models, or the difference between them. I am referring to the mechanisms by which a shock to one variable affects the other variables. Since the baseline NK model does not contain any endogenous persistence, I'm primarily interested in how these effects play out within one time period. That is: what are all the mechanisms through which a shock propagates through the system and determines a new equilibrium $E^*$ (as opposed to the equilibrium $E$ that would have happened in that time period without the shock.)

So I'm looking for an intuitive paper/article on the internal (propagation) mechanisms in RBC and New Keynesian models.

Note again, that I am interested in a primarily intuitive/economic explanation of these mechanisms, rather than an overly mathematical one since the latter I have already found in textbooks.

If possible, clarifying what exactly you're most interested in might help answers be more on point and useful. Are you interested in the mechanisms that cause a one-period shock to last (and not just immediately dissipate)? Or are you interested in understanding some of the reasons that a shock to one variable (say technological progress) leads to changes in other variables (output, employment, wage, interest rates, etc.)?

In either case, the answer depends somewhat on what flavor of the model you're considering. The reasons that shocks tend to have persistent effects in the models comes from mechanically inserting persistence (by using an AR process for shocks) and/or the interplay between stocks and flows (the most common being capital accumulation and investment).

As for sources, I personally found that lecture notes from graduate macro classes were often some of the most useful resources for understanding internal mechanisms and reasoning (Eric Simms, for example, has a wonderful series of lecture notes that include some outlines of BC facts that models try to explain (including persistence), basic RBC model notes, notes examining extensions to the RBC framework, notes on investment in RBC models, in addition to NK notes and some explaining optimal monetary policy in NK models). In addition to Simms' resources, Ordonez has created a nice, simple set that explain the derivation, which might provide some insight. Additionally, slide 15 in this set describes one example of how propogation can work, though again, that's quite a limited example.

That said, there are several classic papers (such as Resuscitating Real Business Cycles especially section 4) which are useful to keep in mind. As for how others generate persistence, you can look at habit formation in consumption (here, though there are many others, of course), variable capital utilization, or any of the other listed methods on slides 40-46.

Though again, more info on what exactly you're looking for could help me refocus a bit. I hope these at least provide a start to what you're interested in!

EDIT: Given your interest within a single period, I think the key is to keep in mind what the "engine" is behind the RBC model- ultimately, it's all about households maximizing utility. They do this by producing (through the firm side of the RBC model) and by enjoying leisure. The specific relationship between these objects leads households to respond to changes in any single value.

For example, take a technology shock (say a positive one, so now more output can be generated from the same levels of input). There are two competing forces:

The substitution effect, which leads households to "realize" that, by increasing the number of labor hours they commit to, they can get even more output (some of which can be used for consumption, since "output" in this model isn't differentiated, and can therefore be used either for consumption or investment).

The income effect, which dictates that, since leisure is a normal good (as defined by the utility function exogenously plugged in to the model), higher consumption should lead to higher leisure as well.

Generally, models are "calibrated" such that hours worked and output both rise. So this suggests that households value the additional consumption enough so they choose to work more.

In RBC models with investment, positive technology shocks also mean that capital is more productive, which means interest rates rise.

I can write a more thorough overview from my grad macro notes later in the day, but until then, I'd review this set of Simms' notes, which do a decent job explaining where everything comes from, and why variables move the way they do. Alternatively, as this source notes: "Proposed by Kydland and Prescott, the RBC theory rests on the neoclassical concept of rational expectations and constructs on the basis of expected utility maximization. Therefore, it is important to understand the central assumption in the RBC theory: individuals and firms respond to economic events optimally all
the time." Additionally pages 10 onwards might be useful in understanding where this comes from, and this set of notes also takes more time detailing where things come from, and what they mean.

Let me know if those additional resources are helpful at all!


Oligopoly kinked demand graph doesn't make a sense to me Hello, so in class I learned about the kinked demand curve in oligopoly. Our teacher said that under the sticky price, the demand is inelastic. However, I don't understand how the demand can be inelastic when the marginal revenue is still positive.

As the graph notes, the red segment of the demand curve is relatively inelastic, meaning that compared to the blue segment, the red segment of demand is relatively insensitive to price changes. This does not mean that price elasticity anywhere on the red segment is less than 1 (which implies inelasticity in the absolute sense).


Intertemporal Model Macroeconomics Basically, I have the full intertemporal model (the two period model) with the consumer, firm, and government, complete with the investment. And there is this question: Determine an equilibrium path with the case in which workers have a preference shock that reduces the disutility from work $h-l$ in first period. Preferences are back to normal in second period.
Note: $h$ is the number of hours available to work, $l$ is the number of hours spent on leisure, so essentially $h-l$ is the number of hours spent working.

What is this question trying to ask? If I understand the question, I probably will be able to answer.

You just have to write down some utility function that changes in each period, in the first there should be an additional term (shock) a constant multiplicative on the disutility of work that reduces it, and then in the second period this constant becomes one.
A dynamic model solution is a vector of $(l^*_1,l^*_2)$ of the equilibrium hours of work that the consumer chooses according to her preferences, and restrictions.