Is the economic behaviour of a competitive market as a whole the same as a monopoly market? We know that the price-setting mechanics and individual firm's behaviour in a competitive market are different with those of monopoly powers'.

However, the demand curves of both competitive market(as a whole)and monopoly market slope down.

So here's the question: can we view the competitive market as a whole as some kind of monopoly power?

PS: What I'm talking about is purely theoretical. Hence pure theory is ok for my question.

Intuitively, I would say no. The main power that a monopoly has is that the firm can change prices at will, whereas a competitive firm has no power over price.

The competitive market demand curve slopes downward simply because of the law of demand. The typical procedure is to take the price determined by the aggregated demand curve as given and set the individual firm's demand curve equal to that. This seems peculiar (to assume to know the market demand before you know the firm's demand), but given the assumptions of perfect competition  (particularly perfect knowledge), this does make some sense. If every firm has perfect knowledge of demand for their good, then they inherently know the market demand curve. The reason why this is important is because that is how you derive the price taking behavior of competitive firms.

Now that we have determined why price taking behavior exists, we can discuss whether competitive markets (as a whole) can behave like monopolies. The answer to this is that it would be nearly impossible for an aggregate competitive market to behave like a cartel (monopoly). The reason for this is because of the perfectly elastic nature of their demand. Suppose, somehow every firm agreed to increase their price to the exact same amount at the exact same time. Suppose further one firm had the ability to produce enough of the good to meet demand at a certain price. This firm (or any firms like it) would have a huge incentive to lower their price because by lowering their price (even by an infinitesimally small amount), they would instantly capture all of the demand for the good. If just one firm decides to cheat, then the whole market would end up right back at where they started at the original market price. For this reason, it is nearly impossible for an aggregated perfectly competitive market to act like a cartel.


Does this cartoon about the transpacific partnership contain errors? http://economixcomix.com/home/tpp/

Discussion of the cartoon spawned a very argumentative thread on reddit, with as far as I can tell no hint of a consensus.

I'm specifically interested in whether the cartoon has any specific factual errors or economic misconceptions, and if so whether those mistakes are serious or not.

Small mistake #1: What capital is

The author identifies "bonds, real estate deeds, stock certificates" with "capital" (p. 5). But in mainstream economics, capital refers not to these, but to durable goods that can be used to produce other goods and services.

It may be argued that this is a mere quibble, since the author is communicating to a mass audience and in everyday parlance, that is what people think of as capital. Unfortunately, "capital" recurs throughout the comic and is, indeed, a key component of his arguments against free trade. He frequently conflates "capital" in the fuzzy, ill-defined layman sense and "capital" in the more precisely defined sense that is used in academic economics.

For example:

On pp. 5-6, capital is depicted as pieces of paper ("bonds, real estate deeds, stock certificates") that China owns.

Then on p. 12, capital is depicted as businesses that are shipped off on boats from Iceland to Guatemala.

Then on p. 13, "stocks, bonds, and real estate" (which he defined earlier to be capital) are given to some Chinese-looking man on the left, who then gives back in turn $ \$ $ to the grouchy, jowly man in the middle, who then distributes this $ \$ $ either as savings or investment. Investment is depicted as some excavators constructing some buildings. The text below then claims, "People who 'buy capital' are investing their money." This is all terribly wrong and confusing: So what exactly is capital? Is it something that corresponds to the $ \$ $ symbol? Is it "stocks, bonds, and real estate"? Is it excavators or buildings? What is its relation to savings and investment?

As already said above, capital refers to durable goods used to produce other goods and services. Investment in turn refers simply to the purchase of capital.

On p. 21, capital is depicted instead as boxes being shipped off on a boat by some corporate fat-cat displaying his middle finger. It is unclear what is in the boxes (pieces of paper? excavators? office buildings?).

P.S. I am not an expert on trade. This small mistake is merely one I managed to spot. In my opinion there are many more mistakes. But this post is long enough, so I hope someone who is an expert on trade will find it her duty to explain the other mistakes, especially since this comic has for some reason received undeserved attention.


Is my logic on taxation for this question legit? At Fenway Park, home of the Boston Red Sox, seating is limited to 39,000. Hence, the number of tickets issued is fixed at that figure. Seeing a golden opportunity to raise revenue, the City of Boston levies a per ticket of 5 bucks to be paid by the ticket buyer. Boston sports fans, a famously civic-minded lot, dutifully send in the 5 bucks per ticket. Draw a well-labeled graph showing the impact of the tax. On whom does the tax burden fall - the team's owners, the fans, or both? Why?

Now, for this question, my take is that the tax burden fall on the fans, because the fans paid 5 bucks extra on tax, and the government imposes a 5 bucks tax. Hence, it doesn't affect the sellers/team's owners. But from what I've learnt is that tax burden is shared amongst sellers and buyers.

I'm equally confused. If it's shared, then who gets to decide the amount buyers and sellers have to split for the tax burden? (If the amount is not provided?) Thank you.

The burden of taxation is shared among suppliers and demanders according to the price elasticities of supply and demand. The more elastic side carries less of the tax burden.

To understand this, note that the tax effectively increases the price demanders pay and decreases the price suppliers get. Elasticity tells us how demanders and suppliers react to this price change. For demanders this means buying less and for suppliers it means supplying less. If you are more elastic then you react to the tax more (by definition of elasticity), which means you avoid it more.

To illustrate this, imagine a tax on cigarettes. We assume that smokers who are addicted will have to buy cigarettes at any price (as they are addicted). This means they do not react and have 0 elasticity of demand. In that case, imposing a tax on cigarettes (a price increase) will not affect how many cigarettes are bought. So the producers of cigarettes are making just as much profit as before, while the consumers are paying higher prices, hence the consumers carry the full burden of the tax.

You could imagine a similar argument, where the supplier produces less because of the tax, thereby causing prices to increase and hurting the consumer.

In your case the producer is totally inelastic, while consumers have some elasticity and hence carries the burden of the tax (as he cannot react and reduce the number of seats).

Alternatively imagine a tax put on producers, where the producers are inelastic, but consumers are elastic. The producers when trying to push the tax onto the consumers (raise the price) the consumers will buy less (are elastic) and thus reduce the suppliers profits, so the producer cannot push the full tax onto the consumers if he's maximizing profits. He can only push a part of it, which means his profits will partially fall and buyers will pay higher prices. Hence both carry the tax burden. The producer can push less of the tax increase the more elastic the consumers are compared to him.

We can also see this general point graphically. The figures I have for this are when the tax is on the producers so affects the supply, but the result is equivalent to a tax on the consumers affecting demand. The blue region represents the surplus lost by the consumer that is captured as tax revenue as a result of the tax and the red region represents the surplus lost by the producer that is captured as tax revenue as a result of the tax. The grey region is deadweight loss resulting from the tax -- it is surplus lost, but not captured as tax revenue. In the first plot below, supply is relatively inelastic (i.e. not very steep) and demand is quite elastic (i.e. steep). In the second, supply is relatively elastic (i.e. steep) and demand is quite inelastic (i.e. not very steep).


What is the definition of distributionally neutral tax? On the website https://www.core-econ.org/ in the book provided, there is an exercise 19.10. It asks whether poll tax is progressive, regressive or distributionally neutral. To answer this question of course we need to know first what is regressive, progressive and distributionally neutral tax.
First of all a poll tax is a tax levied as a fixed sum on every liable individual.

Progressive tax is a tax in which the tax rate increases as the taxable income increases.
Regressive tax is the opposite, it takes larger percent from low-income earners and  lower percentage from higher income earners.

However I could not find the definition of distributionally neutral tax.

My question: What is the definition of distributionally neutral tax?

It depends on what the writer intended.  Here, just over halfway down section 19.10, is a part with the orange title  Progressive and regressive redistribution with a paragraph saying

When the direct effect of a tax or transfer policy (compared to what would happen in the absence of the policy) is a reduction in inequality, it is called progressive. We have just seen that expenditures are more progressive than taxes. If a policy’s direct effect is a rise in inequality it is called regressive. Policies that are neither progressive nor regressive are called distributionally neutral.

Clicking on the blue bolded words gives the definitions

progressive (policy) An expenditure or transfer that increases the incomes of poorer households by more than richer households, in percentage terms.
distributionally neutral A policy that is neither progressive or regressive so that it does not alter the distribution of income.

The definition of regressive seems to be missing but presumably is something which increases the incomes of poorer households by less than richer households, in percentage terms.  Similarly, presumably all three can also be applied when the policy reduces incomes, with progressive being where the richer households are hit proportionately hardest, regressive being where the poorer households are hit proportionately hardest, and distributionally neutral where all households are hit proportionately the same.  There will be other types, such as those which hit rich and poor proportionately the same but middle incomes differently.
The key to answering the exercise comes from considering the proportionate effects on incomes of a poll tax.
Other people have used distributionally neutral in a different sense: for example Michael Linden discussing a flat percentage tax with a single deductible used it to mean a result with the same distributional effect as the income and payroll taxes it might replace; he meant that the change (rather than the policy itself) should be broadly neutral on the current distributional effect of these taxes


What does it mean when we say an iso curve has no interior? From the paper: "Standard Auctions with Financially Constrained Bidders" - by Che and Gale. The authors describe an isobid curve as the curve that represents payments of the same value in a 2-dimensional plane (wealth and valuation). They state that:

"The isobid curves are negatively sloped and have no interior"

What does it mean that an iso curve has no interior?

From Wikipedia:

  In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. [...] If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.)


Why are some economists remarkably productive? I'm a first year PhD student and I randomly go to different professors' websites to admire their works. Usually an economist has around 20 published papers when he/she reaches professorship. But there are a few outliers, like Andrew Lo, Darrell Duffie, Kenneth Singleton and Paul Samuelson, who have like a few hundred of papers published. Are they simply more productive than others? Is the number of publications (of course above a certain standard of quality) a good indicator of the economist's ability?

A short answer to your question is both yes and no. You have to first define productivity: productivity, in my opinion, cannot simply mean publishing the number of articles, but rather, their impact. A typical way of modelling the impact of authors is by looking at their h-index. There are many other ways of measuring productivity as well- where the articles are published is important as well. The top 5 journals in economics are (in no particular order): Econometrica, AER, QJE, JPE and ReStud. Articles published in these journals are generally of high quality. However, there are many articles that are published in these journals that do not always have the desired impact. Many a time, articles ranked in lower journals have a high impact. Consequently, it is extremely difficult to measure productivity. To me, the h-index is a good measure.


How to find the present value of a perpetuity with a non-constant growth rate? I've got a perpetuity problem where an organization pays out 50 equal valued grants each year in perpetuity, adding an additional 5 grants each year (i.e. 55 in year 2, 60 in year 3, etc.).

We've only been shown how to do perpetuity problems with a constant growth rate (e.g. the payout increases by 5% each year), so I'm not sure how to deal with this one.

Anyone have any insight?

Basically you want to calculate
$$
\sum_{t=0}^{\infty} a_t \cdot d^t
$$
where $d$ is the discount factor and where the value paid yearly increases linearly, that is
$$
a_t = a_0 + b \cdot t.
$$
With some rearrangements
$$
\sum_{t=0}^{\infty} a_t \cdot d^t = \sum_{t=0}^{\infty} a_0 \cdot d^t + \sum_{t=0}^{\infty} b \cdot t \cdot d^t.
$$
The first half of this is a simple geometric sequence like the present value of a perpetuity, so
$$
\sum_{t=0}^{\infty} a_0 \cdot d^t = \frac{a_0}{1-d}.
$$
To calculate the other sum we'll use a trick.
$$
\sum_{t=0}^{\infty} b \cdot t \cdot d^t = b \cdot \sum_{t=1}^{\infty} t \cdot d^t.
$$
Then write
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & d + 2\cdot d^2 + 3\cdot d^3 + 4\cdot d^4 + ...
\end{eqnarray*}
You can rearrange this (because of absolute convergence) to
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & d + 1\cdot d^2 + 1\cdot d^3 + 1\cdot d^4 + ... \\
\\
& & \hskip 11pt + 1\cdot d^2 + 1\cdot d^3 + 1\cdot d^4 + ...\\
\\
& & \hskip 45pt + 1\cdot d^3 + 1\cdot d^4 + ... \\
\\
& & ...
\end{eqnarray*}
Every line in this is a geometric sequence, so
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & \hskip 7pt \frac{d}{1-d} \\
\\
& & + \frac{d^2}{1-d} \\
\\
& & + \frac{d^3}{1-d} \\
\\
& & ...
\end{eqnarray*}
Which is again a geometric sequence, thus
\begin{eqnarray*}
\sum_{t=1}^{\infty} t\cdot d^{t} & = & \frac{d}{(1-d)^2}.
\end{eqnarray*}
So
$$
\sum_{t=0}^{\infty} a_t \cdot d^t =
\sum_{t=0}^{\infty} a_0 \cdot d^t + \sum_{t=0}^{\infty} b \cdot t \cdot d^t =
\frac{a_0}{1-d} + \frac{b \cdot d}{(1-d)^2}.
$$


Labor time and the exchange value (Marx). What about the quality of the raw materials? Karl Marx writes in his book "capital", that the exchange value of a commodity is determined by the necessary average labor required to produce the commodity. The labor is measured in time.

Here's my question. There are different types of wood for the production of furniture. Why is furniture made of high-quality wood usually more expensive, while the amount of labor (measured in time) is independent of the wood quality? According to Marx, a table made of low-quality wood should be as expensive as a table made of high-quality wood (apart from negligible differences in the processing due to the wood quality). But this is obviously wrong.

What would Marx answer to this question?

The example you've picked is slightly complicated, because I think there are two possible reasons for the difference in price.  Both could contribute at the same time, depending on your view of how wood production works in your example.

1. The labour value invested in growing, harvesting and processing better quality wood (prior to furniture manufacture).  It is natural to suspect that higher quality wood might be harder or more time-consuming to grow, otherwise nobody would bother growing low-quality wood.

2. Rent.  Marx wrote at length about how rent is extracted in capitalist economies.  If growing the best wood is a matter of owning the most fertile land, then the landowner would naturally extract that surplus value from the wood production process.  This rent then shows up in higher prices for higher quality wood.


Kuhn-Tucker and optimization (continue) This is a question related to the question: Kuhn-Tucker optimization problem and game theory .The question is: Some cultures emphasize more social interaction more than others. Is there a role for culture in the model?

My solution is:

Maximization of the social function with respect to $t_{12}$ and $t_{21}$

max aln( T - $t_{12}$+ $t_{12}$$t_{21}$) + (1-a)ln (w(T-$t_{12}$)) +aln( T - $t_{21}$+ $t_{12}$$t_{21}$) + (1-a)ln (w(T-$t_{21}$))

and the FOCs are:

a($t_{21}$-1)/(T -$t_{21}$ +$t_{12}$$t_{21}$) + a$t_{21}$/(T -$t_{12}$ +$t_{12}$$t_{21}$)=(1-a)/(T - $t_{21}$)

and

a($t_{12}$)/(T -$t_{21}$ +$t_{12}$$t_{21}$) + a(-1 +$t_{12})$/(T -$t_{12}$ +$t_{12}$$t_{21}$)=(1-a)/(T - $t_{12}$)

But I cannot find the optimal solutions. Please share your ideas with me.

I tried this for a=0.5, T=10, w=2 and I get the following real solutions for t12 and t21:
(1) $t_{12}=t_{21}=2.8165$
(2) $t_{12}=t_{21}=1.1835$

For (1) the function is valued at 3.78821 and for (2) the function is valued at 3.68444.

So it seems that (1) is the solution for the maximum.

If it helps, here is the function and the contour sets at those parameter values.

Function

Contour Set ($t_{12}$ on X axis):

By the way, are your FOCs correct? For the FOC with t12, I have:
$-\frac{(1 - a)}{(T - t_{12})} + \frac{a (-1 + t_{21})}{(T - t_{12} + t_{12} t_{21})} + \frac{a t_{21}}{(
  T - t_{21} + t_{12} t_{21})} = 0$


Interpretation of $x c '(x)$ Consider a cost function that is continuous, differentiable and (possibly) convex: $c(x):\mathbb{R}^+\to \mathbb{R}$. I was wondering if there is a "common" way to interpret the expression:
$$xc'(x)$$
in terms of economic intuitions. Any suggestions or references?

If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,
$$ x c'(x) = k c(x).$$
That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is homogeneous of degree 1, then $x c'(x) = ax = c(x)$).
Without homogeneity, I am not sure there is any insightful interpretation you can give to $x c'(x)$, but others may have better ideas. The reason I am pessimistic is the "most natural" way for $x c'(x)$ to appear is in
$$[xc(x)]' = xc'(x) + c(x),$$
which gives
$$\frac{[xc(x)]'}{c(x)} = xc'(x).$$
However, it's not clear to me how to interpret $\frac{[xc(x)]'}{c(x)}$ either. Why should anyone care about $xc(x)$ in the first place? Provided $c(x)$ is the total cost (and not the per unit cost), why would you want multiply your total cost by the number of units you are interested in (what does "the total cost of producing 5 units times 5" means?).
If you let $c(x)$ be the per unit cost, maybe you can make some interpretative sense of  $\frac{[xc(x)]'}{c(x)}$ as $xc(x)$ then becomes the total cost?


How does one asses the impact of a proposed policy? My State government is considering a new regulatory policy at the moment. We need to put together an assessment of the (likely) economic impacts of the policy. How do we do that?

I don't work for the state, I'm an outside volunteer.
I don't have any specific economics education.
I'm hoping to be able to put something meaningful together with just a couple of evenings of work.
There are a few other people whom I'm hoping will help.
At this point even knowing what to google could be a big step forward

The desired end result, I'm pretty sure, is a document saying "Given this trusted data, and these reasonable starting assumptions, and this comprehensible mathematical model, we expect to see about this many more or fewer jobs and about this much more or less business investment". Not knowing any standardized techniques, this feels like trying to forecast the effects of a river dam on next year's hurricanes.

There are a few components to the proposed rules, but they can be considered in isolation. The most important one would require (of large producers) a set rate of reduction of CO2 emissions, which currently have few if any regulations in our state. So, for example, opponents of the rules will want a forecast based on the assumption that activities which are currently emitting CO2 simply shut down instead of migrating to different technology; proponents will want a forecast based on the assumption that wind an solar installations ramp up perfectly.

I'm new here; feel free to edit the tags or question!

Being an economist, I'd say that carrying out a quantitative forecast ("the policy will create/cost x jobs") would require setting up a model, feeding appropriate data and applying appropriate estimations. You would require decent data on the firm level and a detailed model dealing with lots of production factors. In other words, lots of work and probably way beyond your team's resources.
What might be feasible is to do a research on the relevant scientific literature in that realm. In particular, you should look for theoretical work on the expected qualitative effects from such a policy. Second, you may look for empirical studies on comparable policies in other countries and contexts. This might give you a rough picture of what one can expect. Assessing the external empirical results might be hard if you're not an expert in the field, but the following guidelines might be of help:

Prefer authors from academic institutions over e.g. international organizations or (way worse) lobby groups.
Prefer peer-reviewed publications to non peer-reviewed publications. Use Google Scholar as search engine. For any research item, click on "all versions". If the item is published in some scientific journal (say, Journal of Environmental Economics), this might give you an indication for quality.
prefer more recent studies and studies that deal with countries similiar to yours over others.


Why can’t we know the specific outcome of an exchange economy with initial resource endowment? In a 2-person exchange economy we define the contract curve as the subset of all Pareto-efficient outcomes within the ‘lens’ bounded by both agents’ indifference curves through $\vec{R}$ which is the initial endowment vector. It is claimed that we cannot know which point on the contract curve is the actual outcome, as it depends on each of the agents’ bargaining skills. I read this on the Wikipedia page about Contract curves.

But isn’t the solution to the optimisation problem a unique price ratio $\frac{p_1}{p_2}$ (assuming convexity of preferences, local non-satiation and WARP), giving rise to a unique line in the Edgeworth box as each of the consumers’ “budget constraint”, and hence a unique equilibrium which is the intersection of this line and the contract curve?

The idea of the contract curve is just restricting the Pareto set so that no one player is worse off than the initial allocation. There's no concept of "price" involved here. When your endowment is $(x, y)$ and you're asked whether you'd like to get another bundle, $(x', y')$ instead, you only check if your utility is higher. If it is, then you'd take it.

The concept of price comes in with the Walrasian equilibrium (p. 654 of MWG) which is yet another level of refinement on the contract curve (i.e. the "core").


Return on Assets vs Return on Investment 1) Is Return on Investment same as Return on Assets?
There are so many different definition in books so I am now confused...

2) Is ROA = Profit before taxes/Total Assets or Profit after taxes/Total Assets

3) I know to find ROI for some investment or project, but is it possible to find ROI for company from income statement and balance sheet?

No.

Return on assets = net income/assets.

ROI = an informal term roughly corresponding to return on starting capital, or net income/starting capital. Starting capital is any money invested in the business.

Let's say you take out a 500k mortgage on a home with a 50k downpayment. That means 550k of assets. Net of everything you make a $10000 profit from rental income or whatever.

Your ROA is 10k/550k because even though you don't own the home free and clear, the house is still an asset.

ROI is a more informal term comparing how much you invested versus the return. In this example ROI is 10k/50k because you only paid 50k for the home.

ROA uses net income, which means after tax.


What are alternatives to lottery that would create same economic impact? Lotteries can be detrimental to a subset of society since people who often play dont understand stats and that for most part lottery is negative expected value. However, taxes (and portion of ticket sales) are used for funding local projects which can be beneficial to community. Are there any alternative ways to raise this funding besides taxes or taking advantage of people who play the lottery?

Municipal bonds are a method of local funding. The bonds could be funded through borrowing money or tolls. The bonds will probably yield better returns than the lottery for money givers. This means that municipal bonds could be an effective way of generating revenue in a short period of time. On the other hand, one key difference between bonds and the lottery is that at some point, additional funding is needed for bonds, whether through tolls or borrowing.


What's a "belief" in Bayesian games? My book defines player $i$s 'belief' about player $j$s type as $p_i(t_j)$.

It then goes on to say that if $t_j$ is presumed to be uniformly distributed on $[0,x]$, then $p_i(t_j) = 1/x$ for all $t_j$.

That makes no sense to me. What do they mean by $1/x$? So if $x = 1$, then $p_i(t_j = 2) = 1$? Even though $2 \notin T_j$??

$p_i (t_j) $ is the PDF of $i$'s belief over $j$'s types.

Recall that if $t_j$ is uniformly distributed on $[0,x]$, then the probability that $t_j \in [a,b]$, where $0 \leq a \leq b \leq x$ is given by

$$ \Pr ( a \leq t_j \leq b) = \int_a^b p_i (t_j) d t_j $$

In particular,

$$ \Pr (t_j = a) = 0 $$

for any $a$.


Annual data VS Monthly data VS Quarterly data for a VAR model I have read blogs posts that say one should use monthly, quarterly or annual data depending on whether you want to predict monthly, quarterly or annual outcome respectively.
So I guess the same applies to studying the dynamics of two or more variables using a VAR model. For example, the response of $y_t$ to a shock in $x_t$ should be expected to be different at monthly, quarterly and annual frequencies, right? It appears so in my experiment in the attached figure.
So saying that "the response of $y_t$ to a shock in $x_t$ is positive" is incomplete, I guess. Perhaps, it is better to say, "the response of $y_t$ to a shock in $x_t$ is positive at annual frequency, negative at quarterly frequency, and so on".
Or it is that the results should be similar irrespective of the frequency of the data?

For example, the response of $y_t$ to a shock in $x_t$ should be expected to be different at monthly, quarterly and annual frequencies, right?

Generally (and usually), yes.

Or it is that the results should be similar irrespective of the frequency of the data?

Generally (and usually), no. The answer to your question depends on the underlying data generating process (DGP). You can take an example process and analyze it at different levels of time aggregation analytically to figure out implications for what happens when we look at it using different frequencies. E.g. take a monthly AR(1) process
$$
y_t=\varphi_1 y_{t-1}+\varepsilon_t
$$
and see how it behaves when aggregated to quarterly or annually:
$$
\sum_{i=1}^{3}y_{t-i+1}=f\left(\sum_{i=1}^{3}y_{t-3-i+1}\right)+u_t
$$
and
$$
\sum_{i=1}^{12}y_{t-i+1}=g\left(\sum_{i=1}^{12}y_{t-12-i+1}\right)+v_t
$$
where you need to figure out $f(\cdot)$ and $g(\cdot)$ based on the original AR(1) model.


Rationalizabilitiy and Strict Dominance When is the set of rationalizable strategies not equal to the set that is left after IESDS?
My thoughts are:
I know that a rationalizable strategy is that which is a best response for Player i given what Player -i does and thus the set of rationalizable strategies would include all the strategies of player i such that each of them is a BR to any other strategy of Player-i. As such, for sure we will include in the set all of the strictly dominant strategies so if there are only strictly dominant and strictly dominated strategies the set after IESDS will result in the same as the set of rationalizable strategies.
Nevertheless, if there exists also some weakly dominant strategies, the set of rationalizable strategies will also contain this because a weakly dominant strategy is a best response to some strategy -i as it pays more in at least some case. In this sense, the set of rationalizable strategies can comprehend the result not only the IESDS but include the weakly dominant strategies and as such, be wider than the set remaining after IESDS.
I know the last is incorrect because the set of rationalizable strategies cannot be greater than the set surviving IESDS. Can someone clarify the matter?

This depends on the kinds of beliefs players can have about their opponents’ play. If opponents' strategies can be correlated, then the strategies surviving IESDS are exactly the rationalizable ones. But if you only allow opponents to choose independently (this is the standard approach in most textbooks), then these two sets are identical for 2-player games but need not be identical for games with more than 2 players.

Here is an example: Player 3's strategy Z is undominated, but not rationalizable (assuming independent mixtures of players 1 and 2).


Nominal GDP versus PPP in comparing military spending of different countries? Military spending as percentage of GDP is the most used measure of the economic capability of a certain country/military. But Some countries like India for example, have a huge difference between their nominal and PPP GDP (2 billion vs 7.7 billion). So which number is more meaningful to express the economic capability of a certain military ?

One argument for using the exchange rate:

  Market exchange rates are determined by the supply and demand of
  currencies used in international transactions. However, the prices of
  many goods and services on domestic markets are determined in partial
  or complete isolation from the rest of the world. Therefore, the MERs
  do not always accurately reflect differences in price levels between
  countries.

  An alternative is to use purchasing power parity (PPP) conversion
  factors (or PPP exchange rates). The PPP dollar rate of a country's
  currency is defined by the World Bank as 'the number of units of a
  country's currency required to buy the same amount of goods and
  services in the domestic market as a U.S. dollar would buy in the
  United States'. >1  The only PPP rates available for all countries are
  GDP- based basket of goods and services that are major components of
  the gross domestic product. Such GDP-based PPP rates are designed to
  control for differences in price levels and thus to provide a measure
  of the real purchasing power of the GDP of each country.

  Using GDP-based PPP rates instead of MERs for currency conversion
  results in much higher output and expenditure figures for many
  developing and transition countries since they have relatively low
  prices for non-traded goods and services—thus giving the currency
  higher purchasing power. A unit of local currency therefore has
  greater purchasing power within a developing country (which is better
  reflected by using PPP rates) than it has internationally (which is
  what is reflected by using MERs). For those such developing and
  transition countries for whom data was available for 2008, the median
  increase in military expenditure figures from using PPP rates instead
  of MERs was around a factor of 2. Three-quarters of these countries
  would see their relative figures increase by at least two-thirds.
  Meanwhile, using PPP rates would cause the GDP and military
  expenditure figures of most 'developed' countries to fall relative to
  the USA, by a median rate of 17 per cent—reflecting the low value of
  the US dollar at market exchange rates in 2008. However, the
  reliability of such PPP rates is lower than for MERs, since PPP rates
  are statistical estimates, calculated on the basis of collected price
  data for a basket of goods and services for benchmark years. Between
  benchmark years, the PPP rates are extrapolated forward using ratios
  of prices indexes, either GDP deflators or consumer price indexes.
  Like all statistical estimates they are subject to a margin of error.

  Furthermore, GDP-based PPP rates are of limited relevance for the
  conversion of military expenditure data into US dollars. Such PPP
  rates are designed to reflect the purchasing power for goods and
  services that are representative of spending patterns in each country,
  that is, primarily for civilian goods and services. Military
  expenditure is used to purchase a number of goods and services which
  are not typical of national consumption patterns. For example, the
  price of conscripts can be assumed to be lower than the price of a
  typical basket of goods and services, while the prices of advanced
  weapon systems and of their maintenance and repair services can be
  assumed to be much higher. The extent to which this data reflects the
  amount of military goods and services that the military budget can buy
  is not known. Due to these uncertainties, SIPRI uses market exchange
  rates to convert military expenditure data into US dollars, despite
  their limitations.

Stockholm International Peace Research Institute: Monitoring Military Expenditures

But wages are a big part of military costs and those are not paid with PPP GDP but rather domestic GDP. MEASURING HARD POWER: CHINA’S ECONOMIC GROWTH AND MILITARY CAPACITY says that both methods have merits but at least in China, both methods understate military capacity.

  The most appropriate way to compare real military capacity across
  countries is to deflate each country’s actual, or potential, military
  spending by the price of real military services in each country.
  Unfortunately military price indices do not exist for most countries.
  The aim of this paper, therefore, is to develop a simple method for
  computing a relative military price index, that deflates nominal
  spending into units of real military services, using readily
  accessible data. The ratio of two countries’ military price indices
  then gives an exchange rate that provides an index of relative real
  military capacity across countries. We use this military exchange rate
  to compare the level and growth of China’s real military capacity
  relative to the USA. 4 We obtain two key results. First we find that
  the value of the RMB, in terms of its ability to purchase real
  military services, is greater than both its market exchange rate value
  and its PPP value. This suggests that conventional estimates of GDP
  understate China’s real military capacity, but also that PPP exchange
  rate comparisons are much closer to the actual value than market
  exchange rate comparisons. This is mainly because of the very low cost
  of labor in China relative to the USA.
  ...

  If we wish to infer
  country’s miliary capacity based on the size of its economy, however,
  standard PPP indices also pose a problem insofar as they reflect the
  relative price of an average basket of goods produced in the economy
  and this average price index may differ substantially from the price
  of military services. Thus Crane et al (2005) argue that, while
  military services have large personnel costs, a substantial share of
  military equipment purchased by developing-country militaries is
  imported or incorporates com- ponents that are manufactured from
  materials and parts sold at world market prices, such as electronics,
  diesel engines, or aircraft frames. Thus they suggest that, for some
  purposes, GDP measured at market exchange rates may give a more
  realistic picture of changes in military capacity (Crane et al 2005,
  pp.16-17). Similarly, the two principle military statistical
  abstracts, The Military Balance published by The International
  Institute for Strategic Studies (IISS), and the Stockholm Interna-
  tional Peace Research Institute’s (SIPRI) Yearbook report relative
  military spending and relative GDP in terms of $US converted at market
  exchange rates and also at PPP ex- change rates. IISS (2012) notes
  that market exchange rates are likely to understate the true level of
  economic resources allocated towards defence since food material and
  hous- ing costs will be lower in China, but remain noncommittal in
  recommending a preferred price index. 8 Thus they often report a
  weighted average of PPP and market exchange rates when discussing
  China’s real military capacity relative to the USA.


Why is devaluing your currency good for exports? I have the typical answer:

"A cheaper currency means exports will go up because it's cheaper to buy from us!"

Yeah, sure, but that's obviously full of holes.

Let's have a simple situation:

We have China (yuan) and America (\$).
America needs 100 "item" from China every year.
One "item" costs 1 yuan.

Now, let's say currently (situation A):

The rate is 1\$:1yuan
That means America pays \$100 every year.

In situation B, we devalue the Yuan:

The rate is 1\$:2yuan
That means America now does one of the following:

Only pays 50\$ and still gets the same 100 items.

China basically just lost 50\$ a year. China loses.

Still pays 100$ but now gets 200 items instead:

China gets the same 100\$ but is now giving out 200 items instead. China loses.

So, how exactly would China ever benefit from devaluing its currency? It doesn't make any sense to me, you're literally just giving more exports, you are not getting anything more in return.

The only reason I could think of, is that they actually do think that, in order to stay competitive, they need to sell 2 items for 1$ instead of only 1 i.e. they do think it should be that cheap and they don't want the floating currency price to interfere with what they think should be the correct price for their products.

The problem lies with your assumptions. Practical world trade doesn't usually work that way. No one ever devalues their currency by half.

This would be a practical assumption:

US buys toys from China - say x units. It also buy toys from Japan - say y units.

Chinese toys cost xx, and Japanese toys cost yy.

Now, the US would start buying more from China (say x + a), by substituting Japanese toys ( now the US buys y - b) with Chinese toys. (The US would never import more without substituting because it won't import more than what it needs, as pointed out by someone else here)

If production is China had slowed down before devaluation, this would result in more production. More jobs, more income for employees. A devaluation is done to boost exports, resulting in more production, resulting in more employment, resulting in an overall increase in economic activity, whether or not it results in a direct increase in the income received through exports (although it usually does)


How does the government launch of Tomahawk missiles against Syria affect the economy and governmental budget? Last night an air strike was launched against Syria of about 60 Tomahawk missiles. At approximately $832,000 apiece, this airstrike cost the US gov't about 49 million dollars. Initially, this sounds like a terrible waste of government funds, but I think I might be missing something here. It seems like the creation of Military hardware as sponsored by the government might be similar to Keynesian spending -- Which is traditionally (At least by Keynesianists) understood to help the economy. The fact that WW2 helped to pull the US out of the great depression certainly supports the principle that war is good for the economy.

That said, is there any reason to believe that governmental use of expendable military hardware won't actually help the economy?

For the purposes of this question, I'm not terribly interested on Moral arguments about why boosting the economy by using military hardware may be a bad thing.

First, we should remember that the government did not particularly spend any money on the air strike, except for incidental stuff like fuel costs.  It already owned those missiles, which were part of a previous government spending program.  Unlike profit-motive organizations that try to be streamlined, the government buys many things it never uses and doesn't necessarily replace things it does use.  This is particularly true of the military.  The US military has enough stockpile to use incredible force in Syria and still not need to change its future spending.

But anyway, let's talk about your question.  Military is, indeed, an example of stimulus-style spending: a government program that employs people and creates demand across the economy, but which does not generate much in the way of goods and services that we would normally count as wealth.  Engaging in war does increase employment and temporarily increase demand.   To be clear, this should not be confused with helping the economy in the long-term.  The Keynsian argument suggests stimulus spending as a way to smooth out short-term trends, but which must be paid for later.  It's not a way to improve the economy overall.

The wealth of a country is measured by the goods and services it produces (or can buy).  Military spending can jump start an economy full of pessimism or in a bad equilibrium, but it clearly shifts resources and people away from wealth-building and is a long-term economic drag.  Think of military spending as similar to hiring people to break rocks or farm weeds, as was done in earlier periods of our history.  It increases employment and but does not add to the nation's overall wealth.  In fact, you can think of military spending as a transfer payment from the wealthy (who pay most of the taxes) to those who benefit from military spending (weapons manufacturers and associated industries).

As you mention, I'm not saying military spending is good, bad, immoral, or moral.  Just saying economically it's a short-term stimulus, long term drag.


Universal Basic Income - Menial Tasks The Universal Basic Income scenario seems to be one where people are free to cultivate their passions. From what I could gather, the main argument for this to be the best case scenario goes something like, if people are free to choose what they would like to do they would choose something which they are passionate about, which in turn would mean that they will gladly work towards becoming better at it, and create a pleasure-productivity feedback loop.

This seems almost universally accepted, and intuitively right.

My question is, who would take care of the menial but necessary tasks? (Elders' carers, garbage man, plumbers, cleaners...)

Some answer that AI development, will give rise to "unconscious" slave robots who will be able to take care of it. How likely is this?

There seems to also be an idea of the possible emergence of "artisanal" markets, in the sense that even if, for instances, robots are able to function as carers for basic tasks, there would be a premium market for "flesh" carers. And people who would choose to fulfill these roles would be highly remunerated.

Are there any studies, papers or polls on this topic?

Pretty much by definition, "necessary" jobs have to be filled. Wages would have to be bid up so that the positions get filled. This implies some form of relative price shock, or inflation. If robots can do those taks, great, but the plausibility of that outcome is left as an exercise to the reader.

The Universal Basic Income can be viewed as a rebranding of the Negative Income Tax idea that was floating around during the Nixon administration. (Milton Friedman was a supporter.) Studies were done at the time, and the problem with the scheme is the effect on income taxes.

My comments here are my interpretation of some of that analysis. One example is the article "The macroeconomics of a negative income tax" by Hyman Minsky (1969), which was republished in "Ending Poverty: Jobs, Not Welfare" (Minsky). Although Minsky is viewed as a heterodox economist, I believe this particular analysis ends up being similar to the consensus view at the time.

In order to hand everyone a non-negligible amount of cash each month, roughly the same amount of taxes will have to be raised. (One could debate about the exact amount of taxes required, but it would presumably be a significant proportion of the transfers, as it would be safe to assume that most of the transfers would be spent. Otherwise, there would be a major inflationary impact from the program.)

Unless the ultra-rich are willing to pay a lot more taxes, marginal tax rates would have to be high, starting at low levels of income. This creates a massive incentive to drop out of the formal economy, creating a death spiral for the tax system (or equivalently, an out-of-control inflation).

Therefore, the real issue is not the effect of artisanal markets, but rather the global interaction between all low-paying jobs and the tax system. For example, if the marginal tax rate for a minimum wage job is close to 50%, why take that job instead of just engaging in barter or under-the-table jobs, and using the UBI money to cover cash expenses?


How Do Economists Quantify Contagion? In the context of international financial crises, what mathematical methods have Economists employed in order to quantify financial contagion?

Note: For a working definition of contagion I'll use the description supplied by Kaminsky, Reinhart & Vegh (2003):

  We refer to contagion as an episode in which there are significant immediate effects in a number of countries following an event—that is, when the consequences are fast and furious and evolve over a matter of hours or days. This “fast and furious” reaction is a contrast to cases in which the initial international reaction to the news is muted.

I believe the single most commonly used metric for contagion is a sharp increase in the correlation of returns in various countries, either within a country (e.g., the correlation of daily returns in the markets for stocks, bonds, and commodities) or across countries (e.g, the correlation of daily returns in the stock markets in Europe, North and South America, Asia, etc).

While correlation is an easily understood measure, and while it's true that pairwise correlations do tend to rise during crisis periods, using correlations in this manner is not without pitfalls. In particular, sampling correlation coefficients will rise during periods of heightened market volatility (and this is a major difference to, say, regression coefficients), irrespective of whether there's a real crisis or not. For more on these pitfalls see, e.g.,

Forbes and Rigobon, 2002, "No Contagion, Only Interdependence: Measuring Stock Market Comovements", Journal of Finance, vol. 57, pp. 2223-2281.

and

Loretan and English, Evaluating changes in correlations during periods of high market volatility, BIS Quarterly Review, 2000, June, pp. 29-36.


Have there been any true "communist" nations as Marx envisioned? Most individuals refer to the former USSR as a "communist" nation, but, in reality, it was only a "socialist" nation with a high level of governmental control in all areas of its citizens' lives.
Have there been any true "communist" nations as Marx envisioned?

(I am not certain of whether I use established English terminology here).

The answer is no, because in a communist state (as described by Marx), there are no markets, especially, markets for production factors. And in none of the countries that implemented a socialistic socio-economic system, was labor "directly socialized" -it continued to be considered as owned by each worker individually, and it was priced, bought and sold. And it doesn't matter how this was done, i.e. of whether the wages were determined through some market mechanism or decreed by the state. All it matters is that labor was still private property.


Where to learn financial modelling concepts and skills? I'm looking to learn more about the financial analysis of a business. Things like unit economics calculation, income statement, free cash flow analysis, a valuation calculation, etc.
Can you recommend a website or book that has simple explanations, formulas for these concepts without lengthy useless explanations?
Thank you!

When I was studying for my bachelors we learned finance from textbook Corporate Finance by
Jonathan Berk. I think thats good entry into learning some basic financial analysis.


Revealed preferance: elementary question When the market prices are Rs 5 per apple and Rs 6 per orange a person
  buys 16 apples and 28 oranges. When the prices are Rs 4.4 per apple
  and Rs 6.4 per orange ,the person buys 20 apples and 25
  oranges.Assuming that utility is derived only from apples and
  oranges,in which situation is the person better off?

My answer: it can't be said because in the first case the second bundle is not affordable and in the second case the first one is not affordable. Therefore, none is revealed preferred to the other?

Is this correct? Please help. I'm trying to understand this concept. Thanks so much in advance.

Yes, your reasoning and your conclusion are correct.


How, with complete markets, do we have a representative agent? I've read several texts stating that with a complete market assumption, there's always a representative agent lurking. How is it that by assuming complete markets, we're able to prove the existence of a representative agent?

What's the connection between both?

Any help would be appreciated.

It depends on what you want from a representative agent. What you get is a representative agent at the given prices. For any two commodities, you have relative prices. This is where market completeness comes in. All you have to do now is find a well-behaved utility function whose marginal rate of substitution between the commodities corresponds to the relative prices. You can even take the utility function to be linear:
$$u(x_1,x_2,\ldots,x_l)=\sum_{i=1}^l x_i/p_i.$$
Of course, the resulting "representative agent" is utterly useless for any comparative statics exercise or any welfare analysis.


Literature on Private Countries Amidst Anarchocapitalist authors such as Murray Rothbard, are there any who discuss the possibility of privately-owned countries, with their own set of private laws, police services and courts?

Murray Rothbard gave a "natural rights" justification for an AnCap society.

David Friedman (son of Milton Friedman) discusses and gives a consequentialist defence for an Anarcho-Capitalist society. It's discussed in his book 'The Machinery of Freedom' and references 14th century Iceland for an approximation of how a society based on private laws and courts work.

Murray Rothbard, however, is still is the best known AnCap theoretician.


Deriving a demand curve from a Cobb-Douglas utility Probably a daft question but I derived an equation for a demand curve from a general Cobb-Douglas utility function $$U(x,y)=\beta x^{\alpha}y^{1-\alpha}$$ given a budget constraint $$M=xP_x+yP_y$$ and found that the quantity of $x$ demanded would be $$x=\frac{\alpha M}{P_x}$$.

which surprised me in that this function is independent of the price of good $y$. I see some references on these pages which suggests that I didn't make any spectacular  errors but, my elementary understanding is that the demand curve is a functions of many thing, not least the price  of other related goods. Now surely goods $x$ and $y$ are substitutes to some extent?

Now, my question is this, is my understanding of this essentially correct and, what sort of utility functions would generate demand curves $x=f(P_x, P_y, M, etc)$?

Thank you

If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over $n$ goods, $x_1,\dots,x_n$) takes the form
\begin{equation}
  u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho},
\end{equation}
where $\rho\in(-\infty,1]\setminus\{0\}$, $\alpha_i\in[0,1]$ and $\sum_i\alpha_i=1$. We interpret $\alpha_i$ as the consumption share of good $i$ and $\sigma\equiv\frac{1}{1-\rho}$ as the constant elasticity of substitution. Note also that when $\sigma=1$ (or $\rho\to0$), we get the Cobb-Douglas utility form.

Solving utility maximization subject to the usual budget constraint, we get the demand for good $i$ as
\begin{equation}
  x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n.
\end{equation}
Again, observe that when $\sigma=1$ we get the demand associated with Cobb-Douglas utility.

The elasticity of substitution governs how relative expenditures on different goods change as relative prices change. Take a two-good example. An increase in the relative price $p_1/p_2$, i.e. good 1 becoming relatively more expensive, causes two effects simultaneously:

per unit expenditure on good 1 rises, as good 1 now costs more in relative terms; and
quantity demanded for good 1 decreases due to law of demand.

These are opposing effects on the expenditure on good 1 relative to that on good 2. It turns out that the elasticity of substitution determines which effect dominates. If $\sigma>1$, the second effect dominates, and if $\sigma<1$, the first effect dominates. When $\sigma=1$, which is the case of Cobb-Douglas, the two effects cancel each other exactly, so that relative expenditure is independent of the relative prices, and depends only on preference parameters (the $\alpha_i$'s).

Of course, CES utility is not the only class of utility functions that generate demands dependent on prices of other goods. Another common form of utility function, the quasi-linear utility function,
\begin{equation}
u(x_1,\dots,x_n)=x_1+v(x_2,\dots,x_n),
\end{equation}
where $v(\cdot)$ is strictly increasing and strictly concave, also generates demand functions that depend on other prices. A common example is $u(x_1,x_2)=x_1+2\sqrt{x_2}$. I trust that you can verify that both $x_1$ and $x_2$ depend on the two prices $p_1$ and $p_2$.


When is a utility representation differentiable? We know that if we start with a connected, separable product space $V_1\times,...,\times V_n$ and a complete, transitive, and continuous preference relation $\succsim$ on this product space, that there exists a continuous! utility representation $u:V_1\times,...,\times V_n\rightarrow \mathbb{R}$. However, continuity $\neq$ differentiability. Thus, I am curious under which conditions there even exists a differentiable utility function.

My first idea would be to at least restrict the product space to $\mathbb{R}^n$, but there may be counterexamples. For example, if $n=1$ and $u$ is the Weierstrass function, $u$ is continuous, but not differentiable.

Since in Economics we constantly work with differentiable utility, I am wondering which assumptions are necessary to ensure differentiability.

A continuous, transitive, complete, convex, and locally non-satiated preference on an open convex subset $V$ of some $\mathbb{R}^l$ has an $r$-times continuously differentiable utility representation with no critical points if and only if the space $I=\{(x,y)\in V\times V\mid x\sim y\}$ is a $C^r$-manifold.

This result is essentially Proposition 2.3.9. in the book "The Theory of General Economic Equilibrium: A Differentiable Approach" by Andreu Mas-Colell. Actually, the result there replaces convexity of $V$ and the preference relation to be represented by the weaker condition that all indifference curves are connected.

I'm not aware of any more elementary approach to the problem.


How to measure GDP in practice? You can find many explanations of theoretically how GDP is measured. But how is it done in practice? Is it estimated by extrapolating from small samples of activity or can the overall value be measured directly somehow.

Beyond what is written in this Economist article, the general principle is an estimation based on statistical data. The 4 major aspects are private consumption, government consumption, investment and net exports.

I've worked as an auditor and cooperated with colleagues from the national bank of Portugal, which collected and cross-checked financial data to be able to calculate the GDP. Looking at the individual components:

The private consumption can be assessed by surveying individuals, making  assumptions in connection with the tax information and financial aggregated data from banks (who are obliged to report to the National Banks)
Government consumption is publicly available
Investments are estimated by looking at company data and company tax filings
Exports and Imports are calculated by crossing the info on  private consumption, investments, and bank data on payments sent overseas


possible Heteroskedasticity? Based on the graph, would you say that there is some sort of heteroscedasticity in the data? Y-axis is residual squared and X-axis is predicted values of Y

The question should be: Does there appear to be enough heteroskedasticity so that not taking it into account would lower the quality of inference? And this is because "taking heteroskedasticity into account" (even if only for robust standard errors) is not without costs -with small sample sizes it may worsen the reliability of results. And it becomes even riskier if you want to implement the more traditional approach where you will specify a functional relationship between the error variances and the regressors.

From the graph, my initial conjecture would be "no, not enough". But peform an "influential observations" analysis, since you obvioulsy have a few large residuals.
Also, with today's software, it is cheap to estimate a heteroskedasiticity robust variance covariance matrix, and see whether standard errors differ visibly or not, whether they reverse statistical sigificance tests or not, etc.


How to prove that a utility function U(x,y)=min(x,2y) is quasiconcave? I have a question that asks:
"Let there be two goods 1 and 2.Let $x$ and $y$ denote their respective quantities.$(x,y)$ represents a bundle. Suppose a consumer’s preferences over bundles in $R^2_+$  can be represented by the utility function $U(x,y)=min(x,2y)$. Prove that the consumer’s preferences over bundles in $R^2_+$ are convex by proving that the utility function is quasiconcave in $R^2_+$"
I was taught that if a utility function is quasiconcave, then $\begin{vmatrix} 0 & U_x & U_y \\ U_x & U_{xx} & U_{xy} \\ U_y & U_{yx} & U_{yy} \\\end{vmatrix} > 0 \forall (x, y) \in R^2_+$
However, when I start doing the partial derivatives, I notice that I get:
$\frac{\partial U}{\partial x} = \begin{cases}1, && x \le 2y \\ 0, && x > 2y\end{cases},$
$\frac{\partial U}{\partial y} = \begin{cases}0, && x \le 2y \\ 2, && x > 2y\end{cases},$
$\frac{\partial U}{\partial x^2} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$
$\frac{\partial U}{\partial y^2} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$
$\frac{\partial U}{\partial xy} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$
$\frac{\partial U}{\partial yx} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases}$
I then got $\begin{vmatrix} 0 & U_x & U_y \\ U_x & U_{xx} & U_{xy} \\ U_y & U_{yx} & U_{yy} \\\end{vmatrix} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases}$
This is where I am stuck. I am unsure if:
a) I made a mistake somewhere
b) my method is wrong
c) Everything I've done so far is correct, but I need to do another step
Please let me know if I need to provide more information, thanks

A function $f:D\rightarrow \mathbb{R}$ is said to be quasiconcave if the following set is a convex set for every value of $a\in\mathbb{R}$:
$P_a = \{x\in D: f(x) \geq a\}$
To show that $f(x,y) =\min(x, 2y)$ is quasiconcave, we just need to show that $P_a = \{(x,y)\in \mathbb{R}^2: \min(x, 2y) \geq a\}$ is a convex set. For that we consider arbitrary $(x', y')$ and $(x'', y'')$ from the set $P_a$ and arbitrary $\lambda\in [0,1]$ and show that
$\lambda (x', y')+(1-\lambda)(x'', y'')$ is in $P_a$. Observe that $x'\geq \min(x', 2y')\geq a$ and $x''\geq \min(x'', 2y'')\geq a$, so $\lambda x'+(1-\lambda)x''\geq a$. Likewise, $\lambda 2y'+(1-\lambda)2y''\geq a$. Therefore, it follows that $\min(\lambda x'+(1-\lambda)x'',2(\lambda y'+(1-\lambda)y'')) \geq a$ and consequently, $\lambda (x',y')+(1-\lambda)(x'', y'')$ is in $P_a$.


Are imperfect-information games in extensive form exactly the same as sequential incomplete-information games? According to Wikipedia, an incomplete-information game can be converted into an imperfect-information game with complete information in extensive form by using the so-called Harsanyi transformation, which means adding chance nodes at the beginning of the game:

It may be the case that a player does not know exactly what the
payoffs of the game are or of what type their opponents are. This sort
of game has incomplete information. In extensive form, it is
represented as a game with complete but imperfect information using
the so-called Harsanyi transformation.

My question is: is this transformation feasible only for some kind of incomplete-information games or for all of them? I easily see how it can be done when the distributions of probabilities over the possible types of players are known. In this case, you add one or more chance nodes at the beginning of the game, labeled with said probabilities. But what if the distributions are not known? Can the transformation be made? In this case, I suppose the chance nodes added would not be labeled. Would this still be an imperfect but complete-information game? Would it still be an extensive-form game at all, considering that in the definition of the extensive-form game given by Wikipedia the probabilities of actions by nature (chance nodes) are known?
If the answer to this question is "only certain kinds of incomplete-information games can be converted in imperfect but complete-information games", then another question naturally rises: considering that there is no difference in modeling an imperfect-information game and some particular kinds of incomplete-information games, are imperfect-information games a subset of incomplete-information games?
If the answer is "all the incomplete-information games can be converted in imperfect but complete-information games", then I would ask if imperfect-information games are exactly all of the games with incomplete-information.

Two points:

An Incomplete-Information Game is not well-defined without specifying the beliefs of the players. This is specially relevant if you want to use the standard Bayesian equilibrium techniques. There is still significant leeway here - you may define subjective probabilities for each players, players may have inconsistent beliefs (look at Yildiz's excellent papers in a bargaining setup), etc. Recently, there have been some papers that deal with situations where a player is uncertain about the beliefs held by other players (see Gab Caroll's Robustness and Linear Contracts for a brilliant exposition)- but you may need to develop additional tools (like updating rules under ambiguity - see Faruk Gul's paper) to study these situations.

As for Harsanyi's transformation, the critical component of his 1967-68 paper is the common prior assumption (so every player has consistent beliefs regarding the uncertain elements of the game. Thus Harsanyi states that any incomplete information game with consistent beliefs can be remodelled as a game on imperfect information by introducing Nature's move at the beginning. There may have been extensions to the result in the past 30 years, but Im not aware of it.


Original data Altman used for his z-score formula At our university we are researching the usefulness of failing (bankrupticy) prediction models. We are interested in the original datasets Altman used for his models. Does anyone know where this datasets can be found?
We have the same question for the datasets Ohlson used.

The original paper (of Altman) is

Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The journal of finance, 23(4), 589-609.

We read (p. 593)

  III. DEVELOPMENT OF THE MODEL

  "Sample Selection. The initial sample is composed of sixty-six
  corporations with thirty-three firms in each of the two groups. The
  bankrupt group (Group 1) are manufacturers that filed a bankruptcy
  petition under Chapter X of the National Bankruptcy Act during the
  period 1946-1965. (...) Recognizing that this group is not completely
  homogeneous, due to industry and size differences, a careful selection
  of non-bankrupt firms was attempted. Group 2 consisted of a paired
  sample of manufacturing firms chosen on a stratified random basis.
  (...) Firms in Group 2 were still in existence in 1966. Also, the data
  collected are from the same years as those compiled for the bankrupt
  firms. For the initial sample test, the data are derived from
  financial statements one reporting period prior to bankruptcy.
  (footnote 18: The data was derived from Moody's Industrial Manuals and
  selected Annual Reports. The average lead time of the financial
  statements was approximately seven and one-half months prior to
  bankruptcy.)

The paper does not contain, even in print form, the actual data sample, so BKay's suggestion to contact the authors is your best shot.


Game theory: Textbooks I am really a beginner guy in game theory.

Which kind of books could you suggest me as beginner/intermediate/advanced textbooks in game theory ?
I have no problem in maths, so any books with full of maths are ok for me.

As I don't really know so much about this tool, I can not evaluate which book fits well to my situation.

My suggestions:

the best ever written advanced book on game theory is 'Game Theory' by Fudenberg and Tirole: it deals with a lot of material used in research and many applications. When you work on a game theory paper and, for instance, u'r unsure about what kind of equilibrium applies to your case, the book will save your life. It's a bible
but if you want to get inside the game theory world, learn about what kind of problems game theory is able to deal wth, or even getting some interesting ideas to work on your own, after learning the necessary tools..in these cases I strongly suggests to use 'Game theory evolving' by Herbert Gintis. He was my professor at the PhD, and I appreciated his ability as educator (other than a researcher)

I hope you'll find it useful.
Enjoy!


Euro BUND Future CTD Bond I'm trying to understand valuation of the Euro BUND Future.

As to my understanding, the price of the future should be:

The price of a government bond issued by the Federal Republic of Germany
That bond must have a maturity date between 8.5 and 10.5 years from expiration of the futures contract
The bond must have an issue volume of at least 5 billion EUR
The price is converted by a conversion factor. That factor depends on the coupon. The conversion factor crosses 1 at a coupon of 6%.

Now, the current price of the September contract of the Euro BUND future (FGBLU19) is (at the time of writing) 172.35 EUR.

Now I looked up bonds that meet the criteria, and found: ISIN DE0001135143 (in German, it's tricky getting quotes for German bonds in English).

Now to my checklist:
 - Maturity date is January 4th, 2030. From September 6th, 2019 (Futures contract expires) that's less than 10.5 years and clearly more than 8.5 years.
 - Issued by Federal Republic of Germany, checks out
 - Issue volume more than 9 billion, fine
 - Coupon is 6.25%, that's even more than 6%. Conversion factor should be even better than the 1 I'm assuming (it should be 1 at 6%)

Now the current price (as of writing) of this bond is 170.59%.

Even without the conversion factor (which should make things even better), that looks to me like an arbitrage of 1.75% (when selling a futures contract and buying the bond, holding till September).

Finding that big of an arbitrage so easily makes me skeptical, so I'm assuming I'm overlooking something. What is it?

Sometimes it helps writing it down, I think I figured it out:

I forgot the interest between September and January
At 6.25%, that's around 2.1% for four months
That roughly matches the discrepancy I've noticed, with the conversion factor that could very well check out

So, problem solved, I guess.


Dec 2015: What would cause natural gas and crude oil to go up? Natural gas prices are down like crazy this year as you can see from the chart below:

I don't understand how these prices are maintainable. I know these this price trend is largely because of the abnormally warm weather this winter. I also noticed crude oil down a ton, and that the coal industry is almost dead.

If prices are down so much why are suppliers even continuing to supply gas when their margins are low and no one is buying it? What would save the industry?

Two things would cause the price to go up.

The supply curve could move to the left: some suppliers could leave the market, some wells might switch off.

The demand curve could move to the right: new sources of demand; e.g. a burst of extremely cold weather.

This price drop is a step towards economic efficiency. You'll know when it's more efficient, because almost the whole oil & gas industry will have closed apart from a few niche applications.


What is the iff condition for a preference with linear Engel curves (all Engal curves are linear)? If we restrict the consumption at $\mathbb R_+^n$, then it seems like we are implicitly assuming that the Engel curve pass through the origin, so the iff condition would be homothetic preference.

However, if we let $x\in\mathbb R^n$, other classes of preference would also give linear Engel curves. For example, those represented by a quasi linear utility in two dimensions: $u(x)=x_1+f(x_2)$; in this case, the Engel curves are parallel lines.

It could be assumed that, as in the standard consumer theory, the preference is non-satiated and continuous, so the indifference sets are curves.

Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.

Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.

Let the line $L$ consist of the points $((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$ where $a,b,c \in \mathbb{R}$. I will denote this set of points by $L$, the line. It is assumed that we are talking about an actual line, so $a,b$ are not simultaneously equal to $0$.

Let
$$
U(x,y) = \left\{\begin{array}{lc}
\arctan (b \cdot x + a \cdot y) & \text{ if } (x,y) \in L \\
-\pi & \text{ if } (x,y) \notin L.
\end{array}
\right.
$$

Claim The utility function will always be maximized by a point on $L$.

This is easy, because $\arctan$ maps to $(-\frac{\pi}{2},\frac{\pi}{2})$, thus its points always yield greater utility than other points. We specified that the slope of $L$ is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on $L$.

We now come back to the question of whether income $I$ can be negative. If they can be, than every point $L$ will be a solution of the consumers utility maximization problem for a certain $I,p_x,p_y$. The reason for this is that $b \cdot x + a \cdot y$ creates an ordering of the points of $L$. (Lines perpendicular to $L$ have a slope of $-b/a$, these are the level curves of $b \cdot x + a \cdot y$.) Select a point $(x_0,y_0) \in L$. By setting $I$ to
$$
I = p_x \cdot x_0 + p_y \cdot y_0,
$$
the consumer has barely enough money to purchase this basket. If $L$ is non-negative sloped and prices are positive then no points with higher utility are attainable.

The construction I gave is not unique. One could also use the continuous utility function
$$
\hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right)
$$
to achieve the same result. (If $a = 0$ one may use
$$
\hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right)
$$
instead.)


Profit for business owners vs. incomes of the self employed I am learning the way to calculate GDP with incomes. So there is labor income and capital income. One element of the labor income is incomes of the self-employed. And for  Capital income, an element is profit for business.

I not too clear on the definition of the self employed. Don't people who are self-employed also own their

I'm guessing you may be confused why we add both wages of self-employed people who run businesses, and profits of businesses, as they should be the same given that profits of businesses go to their owners.

The important fact is that when we calculate the profit of each business, we deduct wages (even to their owners) from their revenues and the left-over is what we add as profit.

Therefore, the wages deducted is added back into the GDP via wages of self-employed people.

Hence, there is no double-counting in the Income Approach to GDP calculations.


Risky Assets over Bernoulli Utility My question is about 6.C.4. of Mas-Colell et.al.'s Microeconomic Theory book.

We have N risky assets with returns $z_n, n = (1,...,N)$ per dollar invested which are distributed over $F(z_1,...,z_n).$ All returns are non-negative with probability one. For an individual with a continuous, increasing, and concave Bernoulli utility function $u(\cdot)$ over $\mathbb{R_+}$, we define the utility function $U(\cdot)$ over $\mathbb{R}_+^N$, the set of nonnegative portfolios as:

$$U(\alpha_1,...,\alpha_N) = \int u(\alpha_1 z_1 + \cdots + \alpha_N z_N)dF(z_1,...,z_N)$$

We are asked to show for part c that $U(\cdot)$ is continuous.

The book's solution is a little difficult (particularly since I haven't really worked with measure theory before), but my teacher has said there is a more straightforward way of doing the question. I have questions about both of these points.

First, looking at the book's argument.

We create a sequence $(\alpha^m)_{m \in N} \to \alpha \in \mathbb{R}_+^N$. Then if $U$ is continuous, we should get $$(\alpha^m)_{m \in \mathbb{N}} \to \alpha \implies U(\alpha^m)_{m \in \mathbb{N}} \to U(\alpha)$$

$\exists \ \delta > 0$ s.t. $\alpha^m \leq (\delta,...,\delta) \ \forall \ m$

Since $U(\delta,...,\delta)$ is finite, $z \to u(\sum_n \delta z_n)$ is integrable.

Since $u(\cdot)$ monotone and returns are nonnegative with prob. zero:

$$u(\sum_n \alpha_n^m z_n) \leq u(\sum_n \delta z_n) \ \forall \ m, (z_1,...,z_N)$$

Since $u(\cdot)$ continuous,

$$u(\sum_n \alpha_n^m z_n) \to u(\sum_n \alpha_n z_n)$$ for almost all $(z_1,...,z_N)$

Here the book decides to apply Lebesgue's dominated convergence theorem. We are in some measure space $(S, \Sigma, \mu)$ which states that  for some sequence of functions ${f_n}$, if it pointwise converges to some function $f$ and

$$\mid f_n(x) \mid \leq g(x) \ \forall n \quad \text{in index}, \ \forall x \in S$$

(recall we constructed $g(x)$ earlier)

Then $F$ is integrable and

$$\lim_{n \to \infty} \int_S \mid f_n - f \ \mid d \mu = 0$$

$$\implies \int_S \mid f_n - f \ \mid d \mu \to f d \mu$$

So

$$\int u(\sum_n \alpha_n^m x_n) dF(x_1,...,x_N) \to \int u(\sum_n \alpha_n x_n) dF(x_1,...,x_N)$$

$$\implies U(\alpha^m)_{m \in \mathbb{N}} \to U(\alpha)$$

  I added in the direct definition of Lebesgue's dominated convergence for my own clarity, and I think I understand the proof, but my question for this part is what is the measure space? I know $S = \mathbb{R}_+^N$, and $\mu = F$, but what is $\Sigma$ supposed to be? I think it's supposed to be a $\sigma$ - algebra, but I have no idea how to construct/find it here. It is also very well possible that I am completely mistaken on what space I am working on, so there's that.

So then I tried to setup a proof for this question on my own. I also set up a sequence.

$(\alpha^m)_{m \in N} \to \alpha \in \mathbb{R}_+^N$. Then I suppose that there is some sort of jump discontinuity in $U$ and show that it leads to a contradiction.

So $(\alpha^m)_{m \in \mathbb{N}} \to \alpha$ and $U(\alpha^m)_{n \in \mathbb{N}} \not \to U(\alpha)$

  My attempts so far to find a satisfactory proof have been fruitless. Any help would be appreciated, whether it's the proof itself or just an outline or some hints on what information to use from the original question to use.

Regarding your first question, the space on which you can apply Lebesgue's theorem is $\mathbb{R}_{+}^{N}$. The relevant $\sigma$-algebra is the Borel $\sigma$-algebra and the integration corresponds to the Lebesgue measure. Formally, with your notation, the function $f_m$ is defined on vectors $x=(x_1,\cdots,x_N)$ by
\begin{equation*}
f_m(x_1,\cdots,x_N) = u(\sum_{n}{\alpha_n^m x_n})
\end{equation*}

Regarding your second question, it seems to me that constructing a counter-example would be difficult and tedious. Proofs of continuity are rarely done that way. You might ask your teacher what kind of argument he has in mind. A possibility that you might wish to explore is to go back to the definition of continuity. Basically, you want to show that:
\begin{equation*}
\forall \alpha, \forall \epsilon>0, \exists \delta>0 \text{ such that } \|\alpha-\beta\|<\delta \Rightarrow \|U(\alpha)-U(\beta)\|<\epsilon
\end{equation*}
You can try to replace $U(\alpha)$ and $U(\beta)$ by their expression and manipulate the resulting integral. But this essentially boils down to proving Lebesgue's convergence theorem again : this solution is therefore not going to be simpler, but it might help you to understand what is behind this result.


Proving the existence of Nash Equilibrium using alternate approaches Most of the standard books/papers/reading materials prove/state the existence of a Nash Equilibrium by appealing to Sperner's Lemma, or to Brouwer's/Kakutani's FPT. However, I've recently come to know that the existence can be proved in other ways, although I wasn't able to find any relevant material regarding the same.

My question is, that is it really possible to prove the existence of a Nash Equilibrium using some other result(s)?

I would really appreciate any kind of help.

It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem follows from the theorem proving the existence of Nash equilibria. So this is a strong theorem.

The rest of this answer has been made before the question has been edited to specifically exclude it. (After the answer was already posted and the OP read it.)

There are several ways the existence of NE's in finite games can be proven.

One can use Brouwer's fixed point theorem.

Or, one can do so using Sperner's lemma, stopping by at Brouwer's as an intermediate step. Sperner, Brouwer and Nash


Examples of the use of Vector Autoregressive Models I am self-learning Vector Autoregressive Models currently, and have practiced on a few datasets. But I wanted to read some actual research papers that use VAR, so that I get an idea of the level of rigour in their use in actual research or the practical sides to be kept in mind to use them.
When I searched "Vector Autoregressive Models" or its variants in Google Scholar, it displays papers about VAR. I wanted to read papers which apply VAR to answer different questions, such as in monetary or fiscal policy etc. maybe.

There is this study from Banque de France using a Bayesian VAR to analyse and forecast macroeconomic indicators, authors are Gergely Ganics and Florens Odendahl.


How long are consumers willing to wait in order to find a bargain? I have (what I believe to be) a microeconomics question that I'd like to explore:

  How long are consumers willing to wait to find a bargain on an item they'd like to purchase?

I believe the question is easy to understand, but I'm unsure of the methodology to collect proper and relevant data to solve the question.

The answer, I'm sure, is a bit complicated. Though I'm not completely certain, I suspect the relationship is such that time, $t$, is a function of the change in price, $\Delta P$, of a given item (i.e.,$\quad f:\Delta P \rightarrow t).$ The price change could be relative or absolute.

There might be a schedule of different coefficients, depending on the price bracket of different items. For example, one class would include items that many households could afford on a weekly basis (video games, books, t-shirts, etc.), while another class would include items that the average household doesn't buy every week or year (refrigerators, cars, overseas vacations, etc.), and another class would include luxurious items that only the top 2-5% percentile of income earners could afford (court-side tickets to a basketball game, a Ferrari, a bottle of Dom Perignon, etc).

So here's my question to the community here: does anyone know of any study or previous work that addresses this question? If not, does anyone know how I could go about researching this topic? I guess "SurveyMonkey" would be one route, but it seems like a daunting task, considering how many variables I might have to account for.

Check this paper out:

The Value of Time Spent in Price-Comparison Shopping: Survey and Experimental Evidence
Author(s): Howard Marmorstein, Dhruv Grewal and Raymond P. H. Fishe
Source: Journal of Consumer Research, Vol. 19, No. 1 (Jun., 1992), pp. 52-61

With a bit of hand-waving, you can say that consumers are willing to search as long as value of time < amount saved.


What impact would falling oil prices have on the natural rate of unemployment? My intuition is that as the price level falls, cost of inputs is falling for suppliers and their output increases. More people are then employed to increase production - correct?

We need to distinguish between theory and the real world.

In theory, anything can happen. You would need a model that incorporates something that resembles an oil price (many standard models just have a single aggregated good). What then happens depends upon what other assumptions you stick into the model. Whether any of these models have any relationship to the real world is unclear. In any event, you would need to fix a particular model, and as a new question about its behaviour. (E.g., “What is the effect of an oil price rise in {Model X}?”)

In the real world, we need to distinguish the measured unemployment rate, and the “natural rate.” There are a number of ways of estimating the natural rate of unemployment, and what happens to them depends on all of the inputs into the estimation procedure.

What really matters in the real world is what happens to the measured unemployment rate (which may flow into the estimates of the “natural unemployment rate”, as many of the estimation procedures resemble low pass filters). I think the answer is a definite “it depends.”

Firstly, it depends on what country you are talking about. For an oil producer, falling oil prices can easily be a disaster for employment. Many oil producing countries have government revenues that are highly sensitive to oil prices, and so they may be forced to cut back spending. However, they generally have financial reserves, and so can handle limited price falls with limited disruption. Therefore, you need to look at each involved country separately.

For an oil consumer, a rapid rise in oil prices is disruptive; the 1970s experience could be used as an example. Consumers have to spend more on energy, making some products and services unviable. Meanwhile, the increased income is flowing to overseas producers. There is a “terms of trade” shock. The economy will eventually adjust to the new pattern of activity that incorporates higher oil prices, but in practice, the adjustment implies higher unemployment in the near run.

If the fall in oil prices is reversing a previous spike, this will presumably undo the previous disruption. However, if prices were previously stable, effects may be less dramatic. The effects on the economy will depend on how consumers react to the fall in energy prices. If they use the energy savings to buy domestically-produced goods and services, the total volume of domestic production rises, and so firms would presumably need to hire extra workers to produce the greater volume of output. However, the energy savings could just lead to higher savings, leading to no change in domestic production, and presumably employment (in the near run, at least).

The key is that falling input prices are generally not enough to cause increased production by itself; the firm needs to sell a greater quantity of goods, or else the extra production would just represent an undesired increase in inventories. (Since firms have to project demand, they could ramp up production (and hence, employment) to meet projected demand, but if that demand is not realised, they would have to reverse that decision.

For a country that is both a consumer and importer of oil (such as the United States) there is a trade-off between the benefit to consumers versus the potential loss of fixed investment in energy production. Therefore, it could go either way; you would need to do a fairly detailed model to get an estimate.


Which think-tanks on economics do you use to keep up with the news? Whenever I have spare time, I like to go to Bruegel site and check their section on publications in macroeconomics. Here I usually find small publications of 20 something pages. The size is just about right. It's not as difficult as some papers, but it has as much content as one. For me it strikes the exact balance between a small article in the newspaper and a scientific paper.

I was wondering which think-tanks, or which other sources for this type of publications, do you use?  And why?

I like also shorter columns, which are based on good academic papers. So, the VOX, CEPR's policy portal is very good. They occasionally take unsolicited columns, but many are based on CEPR or NBER working papers.

Project Syndicate publishes short and interesting op-ed economics articles.


How does explanatory variables in variance affect unconditional variance of GARCH(1,1) I have a question about the unconditional variance of a GARCH process, where exogenous explanatory variables are included in the variance.
The usual GARCH models the variance using: $$\sigma^2_t=\omega+\alpha\cdot\epsilon_{t-1}^2+\beta\cdot\sigma_{t-1}^2$$
The usual GARCH unconditional variance, without additional explanatory variables, is given by:
$$\sigma^2_=\frac{\omega}{1-\alpha-\beta}$$
My question is, if we include an explanatory variable $x_t$ in the variance equation, how does this change the unconditional variance?
If the model is:
$$\sigma^2=\omega+\alpha\cdot\epsilon_{t-1}^2+\beta\cdot\sigma_{t-1}^2+\phi\cdot x_{t-1}$$
My guess would be that on some day $t$, the unconditional variance would change with the previous days value of $x_t$, so something like:
$$\sigma^2_t=\frac{\omega+\phi\cdot E[x_{t-1}|I_{t-1}]}{1-\alpha-\beta}$$
Hope it makes sence!

Your last expression is not correct because, as noted in the comments, you are after the unconditional variance, which is constant in these models. it should be
$$\sigma^2=\frac{\omega+\phi\cdot E(x)}{1-\alpha-\beta}$$
PS: Also, you are missing a $t$-subscript in your penultimate expression.
--
RESPONSE to comment
Using more explicit notation, we assume that the conditional variance is
$${\rm Var}(u_t \mid I_{t-1}) = \omega+\alpha\cdot u_{t-1}^2+\beta\cdot {\rm Var}(u_{t-1} \mid I_{t-2})+\phi\cdot x_{t-1}$$
Taking expectations through out
$$E\big[{\rm Var}(u_t \mid I_{t-1})\big] = \omega+\alpha\cdot E(u_{t-1}^2)+\beta\cdot E\big[{\rm Var}(u_{t-1} \mid I_{t-2})\big]+\phi\cdot E(x_{t-1}) \tag{1}$$
Now, by the Law of Total Variance,
$${\rm Var}(u_t) = E\big[{\rm Var}(u_t \mid I_{t-1})\big] + {\rm{Var}}\big[E(u_t \mid I_{t-1})\big]$$
Under the assumption $E(u_t \mid I_{t-1}) = 0 \implies E(u_t) = 0$, we obtain the relation
$${\rm Var}(u_t) = E\big[{\rm Var}(u_t \mid I_{t-1})\big] \tag{2}$$
and lagging once,
$${\rm Var}(u_{t-1}) = E\big[{\rm Var}(u_{t-1} \mid I_{t-2})\big] \tag{3}$$
Another assumption is that $u$ is homoskedastic unconditionally. This together with $E(u_t)=0$ means
$${\rm Var}(u_t) = {\rm Var}(u_{t-1}) = E(u^2) \tag{4}$$
Finally, another assumption is that the $x$'s are identically distributed over time, so $E(x_{t-1}) = E(x)$.
Using all these into $(1)$ we get
$${\rm Var}(u) = \omega+\alpha\cdot {\rm Var}(u)+\beta\cdot {\rm Var}(u)+\phi\cdot E(x) $$
$$\implies {\rm Var}(u) =\frac{\omega+\phi\cdot E(x)}{1-\alpha-\beta}$$


Why does an increase in jobs in the USA devalue the the Australian dollar, oil, iron, gas, gold, and copper? Why does an increase in the number of jobs in the USA make make the Australian dollar, oil, iron, gas, gold, copper devalue?

When you look at many things in economics it is worth noting that everything is relative. $1 is only worth what it can purchase. The decisions you make are made in the contexts of your next best option (opportunity cost). In this case, the Australian dollar isn't worth any less domestically. The US has just gotten stronger. With the US outlook being quite positive at the moment, its assets and investments are looking more attractive, thus the appreciation of the USD wrt to the AUD (and many other currencies).


Collusion model with imperfect public monitoring It is well known that consumers' intertemporal incentives may play a key role on the potential of firms to sustain collusion. To my knowledge, most previous efforts studying the topic have assumed perfect observability of actions. How would imperfect public monitoring affect the usual results of industrial organization, regarding symmetrical punishments and price wars mechanisms? Would the results hold?

Since Coase (1972) there has been awareness that intertemporal incentives play a key role on determining industry configuration and the extent to which firms may exercise market power. Ausubel and Deneckere (1987), Gul (1987), Dutta et al. (2007) and Schiraldi and Nava (2012) analyzed this in an oligopolic setting, particularly the effect of product durability on the feasibility of sustaining a collusive agreement. Their findings nevertheless contradict Coase's, as they show that the greater the patience of forward-looking rational consumers, the easier it was for a cartel to sustain collusion.

These results, however, asume perfect observability of firms' actions. So, what would happen if we lift this (rather restrictive) assumption? I'm thinking of Dutta et al. (2007)'s model for studying a cartel in a durable goods market and considering what would happen if one would extend it into a stochastic environment, making consumer's turnover rate random, which leads to an imperfect monitoring setting.

You'll get different answers depending on assumptions, so the answer is maybe. Two papers come to mind.

For a continuous time model, see Sannikov and Skrzypacz (2007). They show that collusion is impossible with imperfect monitoring if new information arrives continuously and actions can respond quickly in accordance with new information.

A well known model using discrete time periods is Green and Porter (1984). They show that collusion is possible. Firms can sustain collusion by allowing for the possibility of price wars.


How does domestic demand affect the domestic currency's value If an economy grows, due to a tax reduction for example, how does that affect the exchange rate?

My logic would be that reduced taxes put more money into people's pockets, which they then use to buy more stuff. If all of that stuff were produced domestically, nothing were to happen to the exchange rate. If some of that stuff were to be produced in other countries and imported, those importing the goods will want to exchange the domestic currency for their foreign currency. That would increase demand for foreign currency relative to domestic currency and thereby decrease the value of the domestic currency.

However, as the economy grows other country's producers will want to buy into the economy's market, for which they will need that country's currency and increase demand for it. That will drive the currency up.

Therefore: What effect is stronger in response to domestically motivated economic growth? Is it the depreciating effect of imports or the appreciating effect of new entrants or am I missing something entirely?

What effect is stronger in response to domestically motivated economic
  growth? Is it the depreciating effect of imports or the appreciating
  effect of new entrants or am I missing something entirely?

There's no general answer to that. It depends on the particulars of each economy. Some factors are:

Whether imports outweigh exports or viceversa;
Whether the tax reduction is coupled with spending cuts, tariffs,
foreign debt, etc.;
Taxpayers' general propensity to save the additional "disposable"
income could influence the exchange rate in either direction,
depending on where they deposit their savings.


Homothetic preferences and utility functions I know that if you have homothetic preferences and a utility function that represents it, then this utility function must present constant Marginal Rate of Substitution (MRS). My question is whether the opposite direction of implication is also true, to be very specific, is it true that a utility function that presets constant MRS always represents a homothetic preference?

As noted in the comments, it is not true that homothetic preferences must have constant marginal rates of substitution.

To see this, recall that preferences given by the utility function

$$ u(x,y) = x^\alpha y^{1-\alpha} $$

are homothetic. (More generally, Cobb-Douglas preferences are homothetic.) However, the marginal rate of substitution is

$$ \text{MRS}(x,y) = \frac{\alpha}{1-\alpha}\frac{y}{x}, $$

which is not constant. However, the MRS is homogeneous of degree zero, since

$$ \text{MRS}(\lambda x, \lambda y) = \text{MRS}(x,y). $$

Homogeneity of degree zero of the MRS is a general property of homothetic preferences. This follows from the fact that (continuous) homothetic preferences have a utility representation that is homogeneous of degree one.

Conversely, when the MRS is homogeneous of degree zero, preferences are homothetic. Hence, preferences that exhibit constant MRS are also homothetic. The proof is a little involved. For this, I refer to you lemma $1$ of "Duality and the Structure of Utility Functions" by Lau (1970). (Note that Lau states a different definition of homotheticity than you do. However, the Lau's definition and yours are equivalent when preferences are continuous -- which they must be for the MRS to be well-defined.)