Alternative normalisation to triangular restriction on cointegrating vectors

The issue is not so much normalization as it is identification of the parameters of the co-integration vectors. The estimated co-integration vectors define the co-integration space, but many other sets of vectors span this same space.

In the case where you have a single co-integration vector, normalization is sufficient for identification. This is not the case when you have more than one vector in which case you need to impose restrictions, the most common set of identifying restrictions being the triangular normalization you mention.

Johansen (1995, section 2 theorem 3) provides a set of algebraic conditions which are fairly easy to compute (but tedious to type) to verify that a set of restrictions imposed on the co-integration vectors is identifying.

It is also possible to identify the co-integration vectors by imposing restrictions on the adjustment matrix ($\alpha$). This fairly recent overview by Johansen might be useful.

Reference: Johansen, Søren. "Identifying restrictions of linear equations with applications to simultaneous equations and cointegration." Journal of econometrics 69.1 (1995): 111-132.


Why is airbnb more expensive than priceline?

It should not be at all surprising that different people choose different offers: the short-stay accommodation market (let's say 1-30 nights) is characterised by a great deal of product differentiation and yield management. Airbnb has reciprocity at play, as guests and hosts switch roles. Additionally, hotels and airbnb hosts have different cost structures and different business models.
Furthermore, consumers have different amounts of information: bounded rationality and satisficing are at work. So even if the offers were identical (they're not), and people's preferences were identical (they're not), then people would still make different decisions.
product differentation
Although you always prefer a lower-cost hotel-room to airbnb, some other people will prefer airbnb, even when presented with identical information. That's because they are not identical offers: they have many  different characteristics (flexibility, physical space, presence of host, services offered, location, the P2P feelgood effect, doing business with individuals rather than faceless chains, staying in someone's unique home rather than just another clone room that looks almost identical in most of the world). And different people weight those characteristics differently: no two consumers have identical preferences.
yield management
Different people will also be booking at different times, with varying flexibility. Hotel rooms, (like airline seats), are one of the paragons of yield management, used in many textbooks' explanations of it.
reciprocity
Because with airbnb hosts can become guests, and guests can become hosts; and because reputation plays a major role in the market, there is reciprocity at play: both sides are incentivised to deal fairly with each other, rather than the practice of many hotels of making repeated attempts at getting more and more money from the customer from additional goods and services sold in addition to the basic accommodation, sometimes by using deception in the initial offer.
cost structures
On the supply side, airbnb and hotel rooms have different cost structures: an hotel room that's empty for the night is a wasted asset, and there may be little opportunity to reduce associated costs. Whereas a home without an airbnb guest is still a home, the residents get to have it all to themselves, and don't have to do any additional hosting, laundry, cleaning or tidying.
business models
A hotel's entire income is dependent on it renting its rooms out and selling associated services. For an airbnb host, the airbnb income is usually an additional source of income, a bonus rather than the thing that keeps a roof over the head and puts food on the table. So there is not an urgent and immediate need to find a guest, allowing them to set higher prices and tolerate a lower take-up.


Solow growth model - analytic proof that Inada conditions imply steady state capital is increasing in the savings rate

We want to prove that

$$\frac{n+g+\delta}{s} > f'(k^*)$$

Replace the left hand side with the equivalent from the expression $sf(k^*)=(n+\delta+g)k^*$, and you get:

$$ \frac{f(k^*)}{k^*} > f'(k^*) $$

Cobb-Douglas case

Without loss of generality, assume $$f(k^*) = {k^*}^{\alpha}$$

Then, the above inequality is:

$$ {k^*}^{\alpha-1} > \alpha{k^*}^{\alpha-1} $$

This is:

$$ 1> \alpha $$

This condition represents a production function with decreasing marginal returns to capital, necessary for the Inada conditions to hold.

General case

A concave function has the following property, for any $x$ and $y$ in the domain:

$$f(y) \leq f(x) + f'(x)(y-x) $$

Rearranging, this is:

$$ \frac{f(y)- f(x)}{y-x} \leq f'(x) \hspace{1cm}, \text{for } y>x $$

This is, the derivative of $f(x)$ at point $x$ is no smaller than the slope of the segment between $x$ and $y$. For the case of a strictly concave function (which is needed for the Inada conditions to hold), the inequality is strict ($<$), except in the trivial case of $x = y$.

For further confirmation, apply the above when $y=0$ and $x=k^*$ (notice here $y<x$ so the sign of the above inequality reverses). Since $f(0)=0$, you get:

$$ \frac{f(k^*)}{k^*} > f'(k^*) $$

which is what we wanted to demonstrate.


Mathematical framework for modelling the relationship between price and sales of a product

The main problem of your framework is that you talk about sales but model only demand. Assuming that the supplier is not a monopolist you should take competing suppliers into account, when estimating sales. To do that any real world examples will have to define the extent of the market. Having said that your framework is a pretty good job as a first try!

@3, 4. Classic micro relies on the irrelevance of independent alternatives. Translating this to your framework $a_2 = f(P_2)$, unless $P_1$ directly affects the consumer's budget in $t_2$. For example - if oxygen was more expencive than your income yesterday, you had to borrow to buy it and today will have to repay your debt. As far as such intertemporal concerns may be important, you need to specify the consumer's planning horizon, which will generally depend on the type of good. For example - if durable goods are involved - cars, fridges etc., and I buy today I won't return as a consumer to this market for the next five years no matter what. However if I know (or expect) that the price of cars will come down next month I will wait until it does and buy than. I.e. it can be that $a_1 = f(P_1, P_2)$. I cannot see why $a_1$ should affect $a_2$ or vice versa through a channel different than the price vector unless there are some asymmetric capacity issues in some periods and demand in excess of capacity thus moves back and forth between periods. I am not exactly sure about search frictions but I think that their effect is quite similar with regard to your question. As long as you have simple consumer goods that you buy every month $a_t = f(P_t)$ is okay.

@5 If the oxygen is so expencive that the purchase violates your consumer's life-time budget constraint I am sorry but he is going to have to die. In this case it is obvious that $a_2 = f(P_2)$ only.

@8 There are goods for which the quantity demanded increases when their price goes up. They are called Giffen goods. Apart of that if your consumer from @5. can afford oxygen he will buy. I.e. it can be that $P_1 > P_2$ and $f(P_1) = f(P_2) = 1$.

Demand is usually modeled as a $Q_t = f(P_t$, control variables) + $\epsilon_t$,

where $f$ is deterministic and all randomness comes from the error term, which should be familiar to you from your daily work.


I think the source of your confusion may be due to the inclusion of the linear trend term in your test. If you exclude the linear trend from the test then you will almost certainly find evidence of a unit root. There is actually an open debate as to whether GDP is trend stationary (i.e. stationary once you remove the linear trend) or difference stationary (i.e. stationary once you take a difference of the series). For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=866624

To help you better understand it is possible to illustrate this with a simple model with a lag and a trend (this also generalizes for two lags but it is good to keep things simple). Consider the following model of GDP:

$y_{t}=\alpha+\beta y_{t-1}+\gamma t+e_{t}$

If we subtract $y_{t-1}$ from both sides we get that

$\Delta y_{t}=\alpha+(\beta-1) y_{t-1}+\gamma t+e_{t}$

where the null hypothesis of our unit root test is that $(1-\beta)=0$. The test depends on whether we account for the constant ($\alpha$) and the linear trend ($\gamma$) or not. If the true process has a linear trend but $\beta<1$ then you will not detect a unit root when including a linear trend in the test. On the other hand if you estimate the model without a linear trend and conduct the unit root test without an exogenous linear trend then the unit root test should not reject the null since the estimate for $\beta$ will move towards 1 to try and capture the linear trend. For this same reason if you have large structural breaks in the data the estimate of $\beta$ will move towards 1 and suggest evidence of a unit root (regardless of an inclusion or a linear trend or not).

Long story short if you include the linear trend in the test even if you exclude it from the estimated regression model then you are unlikely to find evidence of a unit root in an empirical process where a linear trend dominates unless the data were even more non-stationary than it appears.


What major monetary policy actions were taken during the Great Recession (2007-2009)?

First, the interest on reserves and the fed funds rate changes:

Source: FRED

Second, we can roughly see the pattern of the QE actions by looking at the balance sheet of the Fed.

Source: FRED

For details on the timing of quantitative easing actions, see Figure 2 from Fawley and Neely (2013):

Source: Four Stories of Quantitative Easing


Showing the Utility Possibilities Frontier is downward sloping

The first statement does not seem to be true. The UPF exists in the space of utilities whereas the other curves (production, indifference) exist in the product space. As a result, the slope of the UPF may change if a monoton transformation is applied to one of the utility functions whereas the other curves and their slope will definitely not change.


Correct scientific term to describe this system property

The term is "volume discounting", where a supplier offers/accepts a lower unit price in exchange for higher volume of business.
Here, it is the firm/buyer that asks for this volume discount, rather than the suppliers/users that offer it as a commercial policy.

Note that  when the buyer demands a volume discount (when it sets $x$) some conditions must be satisfied in order to be even feasible.
Assume $m$ tasks identical as regards the costs that the user has to incur (time etc). Denote the cost per task $c_a$.

For a single task the user is paid $p$, so its net payoff is $p-c_a$.

Taking on multitasking, the user will be paid in total $m\cdot (p-x)$ and its costs will be $m\cdot c_a$.
In order to take on multitasking it must hold that

$$ m\cdot (p-x-c_a) > p-c_a$$

A necessary but not sufficient condition for this to hold is that

$$p-x-c_a > 0 \implies  x < p -c_a$$

which places an uper bound on $x$, a bound that depends on supplier costs $c_a$. So the firm/buyer that sets $x$  must have a good idea about what $c_a$ might be.

Assume that this holds. Then we need

$$m > \frac {p-c_a}{p-x-c_a}$$

The higher the reduction $x$  in price per task the more tasks a user must take on. This may seem like providing an incentive to the buyer to set the requested $x$ as bigger as possible, but it may be the case that the reduction in unit price is so large that the user cannot take on the necessary number due to physical constraints. Let a maximum possible number of tasks that the agent can hanlde be $\bar m$.

Then we need as a necessary condition

$$\frac {p-c_a}{p-x-c_a} < \bar m$$

so that there exist feasible number of tasks $m$ below the maximum, and above the minimum number required to have higher profitability of the supplier/agent.

Re-arranimng for $x$ we obtain

$$x< \left(1-\frac 1{\bar m}\right)(p-c_a)$$

which is tighter than the previous necessary condition on $x$. So the fimr/buyer must also have a good idea about what the value of $\bar m$ is. Otherwise the whole plan runs the risk of not being adopted by the user.

A situation where the above becomes more feasible is when $c_a$ is not fixed per task but at least initially declines with multi-tasking.


We can define a (positive) cost to be simply a negative benefit.

In which case, a (positive) benefit is simply a negative cost.

So yes, in general, costs can be negative.

Note though that in economics, it doesn't usually make sense to speak of costs in isolation or benefits in isolation. Instead we usually look at the net benefits (i.e. all benefits minus all costs) or net costs (all costs minus all benefits).

So to use your interest rate example, usually when one buys say a 10-year bond, the benefit one gets in return is the positive interest rate. However, countries like Germany and Japan have issued bonds with negative interest rates. So, it seems silly that anyone would buy such bonds -- with negative interest rates, one is clearly getting a (negative) benefit and hence incurring a cost. However, this cost must be compared to other costs -- for example that of safely storing some sum of money. When the proper cost-benefit analysis has been made, we will usually find that for the buyers of these German and Japanese bonds, the net benefits are positive (or equivalently, the net costs are negative).


How can a citizen contribute to get his country out of the poverty trap?

I think the best way for an individual citizen to improve their country's economic development is to raise their productivity, while also following ethical standards. "Raising productivity" is a very traditional economic viewpoint, and it basically requires one to move away from rural or subsistence living and towards activities that contribute to society, as measured by the income that you generate. But, I think there also needs to be a focus on long-term developmental outcomes, which may not necessarily be tied to economic profit because of negative externalities.

I haven't done a lot of research into the lives of poor people, but after reading some chapters from Banerjee and Duflo's Poor Economics, I would say that the economy loses out when people are not persistent, motivated, or when they make economically irrational decisions. For example, poor people in India do not use the most cost-effective treatment for diarrhea (rehydration liquid with salt and sugar) because they do not trust and are not accustomed to Western medicine. Another example is that poor people in Morocco who can't afford food buy televisions, but I understand the reason for this. The reason is that poor people cannot be expected to cut out all luxuries in their lives, because they also need things that make their life bearable and pleasurable. If poor people didn't buy expensive food and other luxury items, the generations below them might rise out of poverty sooner, but such a process takes a long time. Income-earners would not want to forego a better life for themselves just so that their children's children will not live in poverty.

All in all, I advise you to become a change agent in your community. Strive to motivate others to do work. Technology will help, so you will need to try to find out what the most technologically advanced way of doing your work is. That knowledge comes with education, which may be difficult to access in your situation, but nevertheless try to find it. Lastly, public participation in governance is important for economic development, so try to actively participate in your country's governance.


Does non-monotonicity imply non-satiation always? Why or why not?

It depends on the function. A non-monotonic function with satiation:
$$
U(x_1,x_2) = 0.
$$
A non-monotonic function without (global) satiation:
$$
U(x_1,x_2) = \left \lfloor{x_1}\right \rfloor  + \left \lfloor{x_2}\right \rfloor .
$$


Finding data on (estimated) trade cost elasticities of imports

Answering my own question --- although I don't have a a reference per se, I've been told that

if you have no idea at all, -5 is a good ,,first guess``, and
there is a paper (and this is what I don't have a reference for, sadly) where the author worked through all the available estimates and arrived at -4,25 as the ,,best guess``.


UK GDP accounts for profits on sale of home

The $£2000$ appears in GDP in 2000, as production (less intermediate goods consumed) and as value added (the wages of the builders and profit of the developer) and as expenditure (capital formation or investment by the original buyer).

The $£2500$ actually does appear twice in 2001, once positively (by the new buyer) and once negatively (by the seller), as fixed capital formation in the expenditure measure of GDP.

So in aggregate this cancels out and does not add to GDP in 2001 (correctly so, as no value has been added in 2001 - equivalently there has been no production in 2001), but might appear in more detailed elements of the national accounts, for example if the new buyer was an individual while the seller was part of government.


I did try to form a formula like this, is it correct?

No. You need to include the NPV of all years. Right now you are only considering two of those years (and it is not clear which year you refer to in the 2nd term).

Also, it is unclear from the problem whether tuition is paid in the beginning of each year or at the end thereof.

The discounting factor in the 2nd term is wrong. The discounting factor for a cash flow at the end of year $n$ is $$1/(1+x)^n$$.


Is it possible to pay the internal debt of the country and to cover the budget deficit by increasing taxes?

At the time of writing, the question is far too long.

From a simplified accounting perspective:

The government fiscal deficit = all expenditures - taxes.

The change in debt in a year is equal to the fiscal deficit (under simplifying assumptions).

So if the government can raise taxes without affecting anything else, the deficit becomes smaller, or turns into a surplus, in which case the debt falls.

The reason why this is not normally done is that raising taxes reduces growth. The exact effect of tax increases is an object of academic and political debate. If you search for “austerity policies” you ahould be able to find a lot of information.


The Wikipedia article shows it graphically.  But here, you're leaving out the important bit:  "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.

So how does this relate? Let's go bit-by-bit. Refer to the graph on the Wikipedia page as needed.

-"for any bundle of goods" means "choose any bundle $x \in X$, where X is the box in the Wikipedia graph.

-"arbitrarily close" means "choose any $\epsilon > 0, \epsilon \in \mathbb{R}$".  So, pick any non-zero real number as small or as large as you like.

-"there is always another bundle of goods arbitrarily close" means "given $x$ and $\epsilon$, you can always find a bundle $y \in X$ that is within a distance $\epsilon$ from $x%$." This is expressed as $||y-x|| \le \epsilon$.

-"is preferred to it" means that for any utility function $U(x)$, we have $U(y) > U(x)$.

Putting this all together, "non-satiation" means that (without a budget constraint) an agent can "never be satisfied" because no matter what bundle of goods they have ($x$) there is always a $y$ that yields greater utility. "Local" means that $y$ and $x$ can be very similar (ie arbitrarily close to each other in $X$).


Convert a sequential move game into a simultaneous move game

This is basically the idea that any extensive form game has a normal form representation.

http://www.sfu.ca/~shihenl/302/Strategies%20Memo.pdf


Have automation and technological advance been shown to have a positive impact on society and the economy or a negative one?

Would you rather work with a tractor or shovel?

Would you rather use one hand to work or two?...

Mathemathically the formula is:

$$EL  = \mathrm{GDP}_r$$

That is, efficiency ($/h) times hours worked per year equals the real GDP. Same is true for your personal life and personal income. Now, if efficiency decreases, you must either work more or live with less, which is not what anyone wants. Note that even as you increase your efficiency, there is little evidence that such thing decrease someone else's efficiency. So thus the result is income inequality and not "lack of necessities". That means, those with lower IQs generally earn less (since IQ is the primary driver of differences in efficiency between people)...

For the example in question, it seems that efficiency has increased, and at least temporarily the L term has decreased (even though the robot arm builders obviously still need to be employed). However, the cheaper robot arms make it easier for other sectors to accommodate people. So more likely result is a sectoral shift in the labor force, which is what historically has always happened. That is, the factory workers take pay cut, but demand for labor increases in a other sector providing a pay increase and more jobs to that sector. Historically L remains the same but E increases for the whole economy.


Is it easier to borrow money during a recession or high inflation period?

The question wording is somewhat vague, so I am interpreting it as follows: assuming an entity wants to borrow at some fixed interest rate, is it more likely to be able to do so in a recession or in a high inflation period?
The following factors matter.

The willingness of lenders to lend depends up on the state of the cycle. As noted by Henry in a comment, lenders were largely unwilling to extend new loans during the Financial Crisis of 2008. Since defaults generally rise in recessions, lending generally tightens up even under less extreme conditions.
The risk-free curve generally falls in a recession (for countries that control their currency, like the United States). This would make it easier to get a loan versus a fixed rate. However, credit spreads tend to widen (due to greater default risk), so depending on the credit quality of the borrower, the borrowing rate might go up.
Risk-free rates tend to rise in an inflationary period, which might partially be offset by tighter credit spreads.

These factors pull in different directions, so there is not enough information to answer the question as stated.


symmetry of equilibria with heterogeneous players

the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same.

This sounds like an ex ante (or ex interim or ex post) symmetric equilibrium, depending on the timing of the realization of the types. If both dimensions of the types are realized at the beginning of the game, then I'd go with the term "symmetric equilibrium", which is typically understood to mean the ex ante sort.
I think you may be confusing between "strategy" and "action", which may be why you rejected the term "symmetric equilibrium". In the context of Bayesian games, a strategy is a function of types while an action is a particular value of that function. For example, in a first-price sealed bid auction with uniformly distributed private values (types), the symmetric Bayesian Nash equilibrium involves each player bidding according to the same strategy function $s(v_i)=\frac{n-1}{n}v_i$, where $n$ is the number of players and $v_i$ is player $i$'s value of the auctioned object, even though the actual bids will differ based on each player's realized type $v_i$. Nevertheless, players with the same value will place the same bid.


How is the rest of the world making the US an interest free loan by holding US currency?

Think about what a loan is in terms of purchasing power.  You give a bank your signature on a loan contract and they give you something that will allow you to buy real stuff (houses, cars, whatever).  You enjoy this real stuff for some time. Then at the end you have to give the bank something that will allow them to buy real stuff, only more of it.  I'm calling it "something" instead of "dollars" for a reason I will explain below.

In the foreign exchange case, the US gives other countries pieces of paper, which are essentially IOUs.  They give the US either real stuff, which is then enjoyed, or they give the US their currency, which is immediately spent on real stuff.  The US doesn't hoard their currency.  Time passes and at the end they give back the IOU's in exchange for some real stuff.  Only instead of giving back more real stuff than was originally given, the US gives back less (because of inflation).

I called dollars "something" in the loan example and "IOUs" in the macro example because dollars take on a different role in each case.  In the loan, the dollars represent real purchases that are made immediately. In the macro example, the dollars take the place of the loan contract.  They don't buy any real stuff until the end of the transaction.

You can think of it another way by ignoring the dollars.  Foreigners give the US electronics and clothes and stuff in exchange for nothing.  After a while the US gives them back food and machines or whatever, only what the US gives back is worth less than the electronics and clothes it was given.  That's a zero interest (or negative interest) loan.

By the way, it's not just foreigners.  Whoever holds currency has given up something valuable (time, materials, land, etc.) in exchange for paper.  As long as that paper is under your mattress, you are not consuming and the paper is becoming worth less than it was.  You've made a negative interest loan.


Limit of this sequence // broadness of inputs

I think the appropriate discretisation should be something like
$$C_i=\left[\int_0^i\!c(i)^{1-\alpha}\,di\right]^{\frac{1}{1-\alpha}}=\lim_{I\rightarrow\infty}\left[\sum_{j=0}^{(I-1)i}\frac{1}{I}c\left(\frac{j}{I}\right)^{1-\alpha}\right]^{\frac{1}{1-\alpha}}$$

The below figure illustrates where each term of the sum comes from (for the case of $I=5$)

(The reason the sum goes up to $(I-1)i$ is that we start counting from $j=0$ and want to have $I$ bins in total.)

In your example with $c(i)=1$ and $\alpha=1/2$, this would yield
$$\left[\sum_{j=0}^{(I-1) i}\frac{1}{I}\sqrt{1}\right]^2=\left[I i\frac{1}{I}\sqrt{1}\right]^2=i^2.$$
We can compare this to the true value of $C$: $(\int_0^i\!\sqrt{1}di)^2=i^2$. Thus, this particular example is special because the discrete approximation gives an exact expression for the integral even without taking the limit. The reason for this is that $c(i)=1$ is a constant function so a rectangular approximation of the areas beneath it is perfect for any number/size of Rieman rectangles.

But suppose we take $c(i)=i^2$. We then have $(\int_0^i\!\sqrt{i^2}di)^2=i^4/4$ and $$\left[\sum_{j=0}^{(I-1) i}\frac{1}{I}\sqrt{\frac{j^2}{I^2}}\right]^2=\frac{(I-1)^2 i^2 (1-i+I i)^2}{4 I^4}.$$
In the limit this approximation converges on the true value: $$\lim_{I\rightarrow\infty}\frac{(I-1)^2 i^2 (1-i+I i)^2}{4 I^4}=\frac{i^4}{4},$$
but for finite $I$ the approximation is imperfect.


How do we define an efficient tax in microeconomics?

When there is a deadweight loss associated with a particular policy (e.g. a tax) the policy is said to be inefficient under the Pareto Efficiency definition. In the above circumstance, the capital gains tax could have an equivalent increase in government revenue as a lump-sum income tax but would cause a larger decrease in the welfare of those saving/investing. This implies that there is a way to make at least one party better off without making another worse off. This is what the differential between EV and tax revenue is capturing.

The tax is efficient $\iff$ $DWL=0$

under the Pareto Efficiency defintion.


In today's money, what was the value of a 1492 Spanish maravedí?

Walsh (1931):

The author of this work has translated maravedis into dollars of 1929 by reference to statistics on purchasing power in wheat, corn and other staples. Hence his opinion that the maravedi was worth about two American cents of 1929.

According to the BLS inflation calculator, \$1 in 1929 is about \$15 today (2020).
So, if we accept Walsh's estimate, then one maravedi in 1492 converts to about \$0.30 today.
(And if we also accept Satava's estimate, then Columbus's voyage cost 1,765,734 $\times$ \$0.30 $\approx$ \$529,720 in today's USD.)


Seemingly contradictory relationship between bond yields and economic growth?

(Note: there were similar questions in the past, but I did not see an exact match.)
As used in this question, “interest rates” are the rate of interest associated with a policy rate or a deposit. A “yield” is the the rate of return on a bond given its purchase price. For a bond, since the payments are fixed, the yield moves inversely with price.
If we simplify to the case of a zero coupon bond that matures at time T (starting at $t=0$), the price is given as:
$$
P(T) = e^{- \int_0^T f(t)dt},
$$
where $f(t)$ is the instantaneous forward rate. (Based on Rebanato’s “Interest-Rate Option Models”, page 6. Should be in any fixed income textbook.)
Put into words: the price of the bond is given by “averaging” the forward rates over the life of the bond. Higher forward rates implies a lower price (and hence, a higher yield).
The central bank “risk-free” rates for short maturities, and the instantaneous forward rate initially coincides with that rate. Further along the time axis, the forward rate is normally interpreted as the expected policy rate plus a term premium.
If economic growth is expected to be strong, market participants normally expect rate hikes. For example, that is the implied behaviour of “The Taylor Rule,” which the reader can easily find references for.
In a recession, it is entirely possible for the central bank to lower the policy rate, and market participants to expect a strong recovery, and forward rates rise. However, one can look at bond data to see that the policy rate and bond yields are roughly “correlated.”
As for the relationship to equity prices (which show up in the question), the situation is muddled and should be a separate question.


Transition Matrix: Discrete -> Continuous Time

Suppose $B$ is an $n\times n$ matrix of Poisson transition rates, where $B_{ij}\geq 0$ for $i\neq j$ denotes the rate at which state $i$ transitions to state $j$, and $B_{ii}\leq 0$ gives the rate at which state $i$ transitions to all other states. Each row of $B$ sums to 0.

Then if $p(t)$ denotes the probability distribution at time $t$, by definition of $B$ we have the ODE
$$\dot{p}(t) = Bp(t)$$
We know what the solution to this kind of ODE looks like: $p(t)=e^{Bt}p(0)$, where $e^{Bt}$ is the matrix exponential of $Bt$. So, if we want $B$ to generate the Markov transition matrix $A$ after $t=1$, we need to have $e^B=A$.

In principle, to get $B$, we need to invert the matrix exponential, taking the matrix logarithm of $A$. The problem is that each matrix has many matrix logarithms - the logarithm in one-dimensional complex space has infinitely many branches, and this is compounded when we're talking about matrices in $n$-dimensional space. Most of these logarithms will not be satisfactory Poisson transition matrices: maybe they won't be real, or the entries won't have the right signs. Yet it is possible that more than one of them will be: in some cases there is more than one Poisson $B$ corresponding to a Markov $A$, just as in some cases there is no Poisson $B$ corresponding to $A$. It's messy.

Fortunately, there is a situation where life is relatively simple, and it almost certainly includes your own case: when all the eigenvalues of $A$ are positive, distinct reals. In this case, there is only one logarithm of $A$ that will be real, and it's easy to compute: you just diagonalize the matrix as $A=V\Sigma V^{-1}$ and take the real logarithm of the eigenvalues, getting $B=V\Omega V^{-1}$, where $\omega_{ii} = \log(\sigma_{ii})$. Indeed, you don't need to do this yourself: if you use the command $\text{logm}(A)$ in Matlab (presumably Python too), it will give you precisely this $B$.

Given this $B$, all you have to do is to verify that it's actually a Poisson matrix. The first requirement, that the rows all sum to zero, is satisfied automatically due to the construction of $B$.** The second requirement, that the diagonal elements are negative and the off-diagonal elements are positive, does not always hold (I think), but it's easy for you to check.

To see this in action, I'll consider an $A$ for a 3-state Markov process that resembles a discretized AR(1).
$$A = \begin{pmatrix}0.5 & 0.4 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.4 & 0.5\end{pmatrix}$$
Now, if I type $B=\text{logm}(A)$ into Matlab, I get
$$B = \begin{pmatrix}-0.86 & 0.80 & 0.06 \\ 0.40 & -0.80 & 0.40 \\ 0.06 & 0.80 & -0.86\end{pmatrix}$$
This is indeed a valid Poisson transition matrix, as we can easily check that the rows sum to zero and have the right signs - so this is our answer.

The case with positive eigenvalues is pretty important, since it spans all cases where there is not some kind of oscillatory behavior in the Markov chain (which would require negative or complex eigenvalues), presumably including your discretized AR(1).

More generally, the $\text{logm}$ command on Matlab will give us the principal matrix logarithm, an analogue of the principal scalar logarithm that takes all eigenvalues to have imaginary part between $-\pi$ and $\pi$. The problem is that this is not necessarily the logarithm we want, and by looking at it we might miss a Poisson $B$ that does generate $A$. (That's why the positive eigenvalue case, where we didn't have to worry about this, was so nice.) Still, even in these other cases it can't hurt to try and see if it works.

By the way, this problem of seeing whether there is a $B$ that generates some Markov matrix $A$ has been studied extensively. It is called the embeddability problem: see some overview and references in this excellent survey article by Davies. I'm not an expert on technical aspects of the problem, though; this answer is based more on my own hackish experience and intuition.

I feel obligated to close by seconding ecksc's comment and saying that there might be better, more direct ways to convert a discretely fitted AR(1) into a finite-state continuous time process - rather than just taking the matrix obtained via the Tauchen method and making it continuous. But I don't personally know what that better way is!

**Explanation (though I'm rusty): $A$ has a unique Perron-Frobenius eigenvalue of 1, and since $A$ is stochastic the right eigenvector of this eigenvalue is the unit vector $e$. This is still the right eigenvector, now with an eigenvalue of 0, when we take the matrix logarithm.


The deadweight loss from the monopoly decreases. This is because the deadweight loss comes from the price being too high (higher than the marginal cost), which leads to not enough goods being consumed in equilibrium. Since the subsidy redices the price, the deadweight loss decreases. The subsidy itself does not increase the deadweight loss, because the only thing it does is reduce the price and there are no other effects. One issue may be that additional deadweight loss is caused by the taxes required to finance the subsidy. If these are lump-sum taxes then they do not affect the deadweight loss as they are non-distortionary and do not affect prices.

In fact it is possible to completely remove the deadweight loss with a subsidy. This requires lump sum taxes to be an available tax instrument, so that raising the money for the subsidy doesn't cause deadweight loss itself. This is an old result that goes back to Robinson (1934). However, although she first noticed the result, she did not advocate that this would be a good policy, unless the subsidy was financed by taxing the monopolist's rents. This is because of concerns such a policy would have on income distribution.


Change in the marginal utility of leisure with respect to a change in consumption

The partial derivative $\frac{\partial^2 U(\cdot)}{\partial C\partial L}$ tells you how a change in leisure affects the marginal utility of consumption. Therefore if it is positive it means that the marginal utility of consumption is higher when $L=1$. In that case, the agent plans her life to be able to consume more when she is retired and leisure is bigger. In contrast, if leisure reduces the marginal utility for consumption, $\frac{\partial^2 U(\cdot)}{\partial C\partial L}<0$, then it is better to consume more while working, than when leisure is big.


Is this use of a Law of Large Numbers correct?

Are the $\hat{u}_t^2$ independent? If yes, the result is indeed a version of the law of large numbers due to Kolmogorov, and that deals with independent (but not necessarily identically distributed) processes.

Here is the formal statement. Suppose that $\{u_t\}_{t=1}^{\infty}$ is a sequence of independent real-valued random variables that satisfies
\begin{equation*}
\sum_{t=1}^{\infty}{\dfrac{Var(u_t)}{t^2}}<\infty
\end{equation*}

Then the random variable
\begin{equation*}
\dfrac{1}{n} \sum_{t=1}^{n}{[u_t-\mathbb{E}(u_t)]}
\end{equation*}
converges to 0 almost surely.

Edit: the assumption of independence seems stronger than needed, and the following result (an adapted version of the law of large numbers for non-independent processes) might be useful. If $\sigma_{n,m}=Cov(u_n,u_m)$ exists, and if there is $0< \alpha <1$ and $M \in \mathbb{R}$ such that $\sigma_{n,m} < M \alpha^{|n-m|}$ for all $(n,m)$, then
\begin{equation*}
\mathbb{P}(|\dfrac{1}{n}\sum_{t=1}^{n}u_t- \dfrac{1}{n}\sum_{t=1}^{n}\mathbb{E}(u_t)|>\epsilon] < \dfrac{M}{n \epsilon^2}
\end{equation*}
for any $\epsilon$.


Regression of first differenced log-transformed model

What you describe is very standard in academic econometric practice. It essentially answers what is the elasticity of Y with respect to X identified from changes in X and Y.

Here are a few problems that can be introduced by this process

First-differencing is very similar to $Y_t = \alpha + \beta_0 * Y_{t-1} + \beta_1 * X_{t} + \beta_2 * X_{t-1} + \epsilon$ with the additional restrictions that $\beta_0 = 1$ and $\beta_1 = -\beta_2$
First-differencing can introduce serial correlation in standard errors


Meaning of assumptions regarding continuum of agents in economics models

Economic models often have a finite number of agents. In such models, agents are commonly named using the positive integers (i.e. 'natural numbers'): so the set of agents is $\{1, 2, ..., n\}$.

Other economic models have an infinite number of agents. For example, the number of agents might be assumed to be the same as the number of positive integers. In that case, the set of agents is $\mathbb{N}=\{1, 2, ...\}$. It is said to have a 'cardinality' (or size) of aleph-null.

Finally, some economic models have even more agents. In the examples you mention, each agent is named using a real number in $[0, 1]$; so the set of agents is the set of real numbers in $[0, 1]$. This set is said to have cardinality aleph-one and can be shown to be 'larger' than the set of natural numbers (in the sense of 'larger' defined by Cantor).

Of course, this is not very realistic -- the justification, presumably, is to make the mathematics easier. In particular, it allows for integration (e.g. to find total consumption in the economy) as opposed to summation.


Is there always at most one full-support extreme symmetric equilibrium?

This seems obvious.  If $\sigma_1,\sigma_2$ are in the joint support of $s_1,s_2$ and $\tau$ is not, then (writing $P$ for the first player's payoff) you need
$$P(\tau,s)\le P(\sigma_1,s)=P(\sigma_2,s)$$

where $s$ is a convex combination of the $s_i$.  But if you replace $s$ with $s_1$ or $s_2$, this holds, so it still holds after you average over $s_1$ and $s_2$.

What am I missing?


If interest rate increases and consumption in the present increases, is present consumption a normal good?

First, the definitions:

A normal good is one whose demand rises when income rises.
An inferior good is one whose demand falls when income rises.

I will use $c_1$, $c_2$, and $y$ to denote current consumption, future consumption, and (lifetime) income.

Now, if $r$ rises, then as usual there are two effects, the substitution effect (SE) and the income effect (IE). We are given that the Total Effect (TE) is that $c_1$ increases or equivalently, that TE is positive.

Let us write $$TE=SE+IE>0.$$

We now look at each of the two individual effects:

Substitution Effect (SE). You correctly identified the cause of this effect -- "the price of $c_1$ increases". And so, the SE is that the consumer should substitute out of $c_1$ and into $c_2$. That is, the SE unambiguously says that $c_1 \downarrow$. We write $SE<0$.

However, you then made the mistake of concluding that this shows $c_1$ is inferior. But remember that whether a good is inferior or normal depends on responses to changes in income. So, the SE has nothing to do with whether a good is inferior or normal.

Income Effect (IE). We first note that since $$TE=SE+IE>0 \text{ and } SE<0,$$ it must be that $IE>0$.

Here there are two possibilities. I very much suspect the person writing this question overlooked possibility #2 (in which case the answer should be B) and went only with possibility #1 (answer C).

If the consumer is a lender (or saver), then an increase in $r$ implies an increase in $y$. In which case, $IE>0$ if and only if the good is normal. Hence, the answer is C.
However, if the consumer is a borrower,  then an increase in $r$ implies a decrease in $y$. In which case, $IE>0$ if and only if the good is inferior. Hence, the answer is B.


Inflationary recession vs Deflationary recession

Deflationary recession refers to a recession in which prices are falling. An example was the Great Depression. Below you have a graph of the yearly rate of change in CPI. A negative number means a fall in prices, i.e. deflation. Shaded areas mean recession period according to NBER.

Some argue that deflation was actually a factor contributing towards the great fall in GDP, because the Fed did not act soon enough, keeping the money supply low. The Gold Standard is partly to blame for this inaction of the Fed.

Inflationary recession is when a recession is accompanied with increasing prices. It is commonly known in Macroeconomics as Stagflation. That was the case on the 1973-1975 recession in the US. The graph below shows the yearly rate of change in CPI (again, shaded area represents a recession):

The latter happened mainly because of the 1973's oil crisis where the OAPEC (oil cartel) stopped selling oil to countries supporting Israel (including the US), rising its price.


Is there a game that cannot be represented by a game tree?

Well, for this you need to define what a "finite game" is. The usual way to do so, is to define it as an extensive form game, which makes the whole issue circular. But there are different ways to model extensive form games at different levels of generality. For example, the first definition due to John von Neumann and Oskar Morgenstern made a number of strong restrictions on timing that look somewhat archaic from today's perspective. The subsequent definition by Harold Kuhn significantly generalized the class of possible extensive form games. An implicit restriction in the way Kuhn defined extensive form games was that no player is allowed to move through the same information set twice, an assumption weakened in the Ph.D. thesis of John Isbell. So there are various notions of extensive form games at various levels of generality, including versions that allow for infinite games.


What does "r" represent in the total cost function?

r is the user cost of capital.
User cost refers to the expenses borne by the owner or renter of a capital asset resulting from the use of the asset for a given period of time.
A standard metaphor is that of a machine being worn down over time as it is being used. During the time of use it creates a flow of goods through a production process. The capital gets it value from this flow of goods it can create. Simplify this metaphor to a machine that during three months can produce three pieces of bread - one each month - each sold for 1 dollar. The total value of the machine 3 dollars (ignoring discounting) so if you were to buy the machine you would pay 3 dollars. However if you just want to used it for a month it will only produce 1 piece of bread the value of 1 dollar so one month of machine is just worth 1 dollar.
Things are offcourse very much more complicated but you can google how the user cost of capital is derived.


Argentine default: "Rights upon future offers"

This is a good question! It turns out that exactly as you predict, if you get the transcripts or recordings from the meetings with judge Griesa in NY, some of the representatives of the non-holdouts said they were willing to sign off on the RUFO clause, voiding it. The logic, as you describe is that this would free Argentina to pay the holdouts, letting Argentina continue its payments to the non-holdouts too, without a risk of the debt ballooning suddenly. This would have saved everybody lots of headaches!

Argentina's government at the time did not pursue this option, perhaps because they saw more political gain in being the victims of 'the US empire's courts' than in solving the problem. The lack of professionalism and interest in solving the problem that their executives displayed during this episode suggests this is the main reason.

Alternatively, it's plausible that they did not pursue this option because they thought that default was so terrible to non-holdout investors, that these would be of help in pressuring the holdouts to accept a lower payment from Argentina in negotiations to avoid default.

In any case, neither strategy worked. The government lost popularity and was kicked out of office, and the new government eventually reached an agreement beneficial to the holdouts.

One, more thing: the Rufo clause actually wasn't such a big deal because it was set to expire soon! So, the non-payment appears to have been a big circus for the Argentinian president then to try to score some points with the more manipulable fraction of that country's population.


How can a one currency gain against the other if it is pegged aginst it?

Strictly speaking it was not a peg but a cap: the Swiss National Bank said it would sell unlimited Swiss Francs at the rate of 1.2 CHF : 1 EUR.

You could have bet that the Euro would rise against the Swiss franc by borrowing some Swiss Francs and selling them for Euros, and then taking any profit or loss by unwinding the transaction.

If for example you had bought Euros with Swiss Francs in May 2012 and sold back in May 2013, this might have been profitable (at least before transaction costs), but if you had held past 15 January 2015 you would have made a loss when the cap was withdrawn.


What is the tolerable level for multicollinearity? and possible remedial measures for it?

I cannot post this as a comment but researchers such as Farrar and Glauber (1976) have shown that if VIF ≥ 10, this indicates the presence of multicollinearity between explanatory variables. There is no black and white threshold for this within the literature, and it highly depends on your sources and on the person reviewing your work. Moreover, you have not made any mention of the estimator you are using, but as you may know, multicollinearity makes the OLS estimator is less precise. That being said, have you tried standardizing by subtracting the mean? More information on your model and or your own knowledge on this would be useful to get help, in my view.


Inflation, cause or result of monetary emission?

The problematic part of the statement, is the "because of other reasons not important here" part . In other words: "ignore general equilibrium" -which is an unacceptable statement to make when discussing government policy and actions.

Consider the naive quantity theory of money:

$$PQ = VM \tag{1}$$
$P$ is the price level, $Q$ is output produced (measured in quantity), $M$ is money supply, and $V$ is "velocity of money", an indicator of the "transactions technology" in the economy, how fast money circulates around to settle transactions.

Assume now that we are talking for a "small" country that needs to import basic factors of production like raw materials or energy. "Small" here means "with no market power". Such a country is a price-taker in the international market. More over, substitution possibilities for these factors are usually small to non-existent.

Competitive markets or not, the economy's output will be distributed to factors of production and for our purposes, it doesn't matter whether there will be "capital rents" and "profits", or only capital rents. Use for convenience three factors of production and write

$$PQ = rK + wL + p_fE \tag {2}$$

where $r$ and $w$ are nominal, and $p_fE$ is the nominal cost of imported factors. Denote $s_f$ the foreign exchange rate (units of local currency per one unit of foreign currency), $c_f$ the price of the imported factors in foreign currency, so $p_f = c_fs_f$.
Use this and substitute $(2)$ in $(1)$

$$rK + wL +c_fs_fE= VM \tag{3}$$

If something happens to the international market and $c_f$ goes up to $c_f' > c_f$, this will tend to increase the left hand side. This "something" in the international production factors market does not relate to the level of domestic output $Q$, or to domestic money transactions technology, $V$. More over, at least in the short run, factor substitutions will not happen, wages do not move that easily, and firms will maintain their output level while increasing selling prices, to cover the increased production costs. And since the reasons for the increase affect more-or less the whole economy, it is not that likely that competition will stop firms from doing so: they all want to cover their increased costs, they all know that the cost-rise is general and comes from abroad, so they don't need to actually collude in order to sustain a price increase. "Common knowledge" suffices.

So in order to preserve the equality in $(3)$ it appears that we must have

$$rK + wL +c_f's_fE = VM',  \;\; M' > M\tag{3}$$

You see? This is the phenomenon called "imported inflation". Whatever the reasons were for the price increase (the "not important" reasons), inflation was not caused by the expansion of the money supply (that's indeed true), and what else the government could do than raise the money supply to service the higher nominal level of output?

Of course what the story above does not say, is that foreign factors of production will want a "money" that they accept, and most likely this won't be the local currency of this small country. And by increasing the money supply the exchange rate $s_f$ will suffer (increase), because $s_f = h(M), \;\; h' >0$, increasing in this way further the costs of imported factors in terms of local currency, and making the increase in the money supply equivalent to "shoot oneself in the foot". And this is only one more step towards the road to general equilibrium.

The essence here is that
a) it is trivial that there are many other factors that may tend to affect prices upwardly, except money supply expansion

b) in the presence of these other influences, increasing the money supply is not necessarily the appropriate government response.


Why do falling oil prices take stocks with it?

First of all, the fall in prices does not only concerns oil but most commodities, in particular raw materials such as iron ore, copper, nickel and many other, though it is indeed most noticeable for gas and oil. If you want to see the full extend of this drop in price, you can check out the IMF Primary Commodity Prices and their monthly reports, from which the following chart is extracted:

What are the causes of lower commodity prices?

There is no secret: the demand is falling and the supply increase. The demand is falling mainly because of the slowing growth in China. As China imports less commodities to fuel its industry, a large chunk of the demand on these markets has disapeared, leading to lower prices. But simultaneously, the supply keeps steady or increasing on most of the commodity markets. On the oil market for example, 2015 has been a great year of surge or expected surge: OPEC countries are committed to keep flooding the market, mainly because the Gulf countries can afford lower prices thanks to low production costs, while their competitor might not sustain low prices very long. Furthermore, the ban on Iranian exports is going to be lifted, as well as the ban on US exports. The market anticipate already this formidable future new supply, which drives prices down. The US are quasi independent as far as Oil is concerned, as the Shale gas industry has been rising in this country.

What are the consequences of lower commodity prices?

The first consequence is not really caused by lower commodity prices, but rather by the root of this price drop: China's growth is slowing. Therefore their is more uncertainty than before about this market. As a consequence of this, exporters to China begin to worry as well as firms who have factories in China, as a slowing growth and reduced imports might be the sign of a stabilization and potential lower inner demand. China being one of the world's biggest market, and one with a still relatively low penetration for technologies, it is indeed worrying. Thus, the stock of a lot of companies have been going down all over the world because of the situation in China, reflected in Oil prices.

Another very important factor comes into action: most firms in the commodity sector have huge debt. Indeed, this sector requires big investments to begin the exploitation. Among others, Glencore's $30bn debt has been discussed a lot, but it is only an example among many others. In particular, their are two areas that are under threat: shale gas and renewable. This two sectors require higher investment than others. For shale gas, because the exploitation is costly and technically complex. For the renewable, because the R&D is costly and the production as a whole is costly as it uses high technologies. Lately, the partial default of Abengoa gave us a taste of what might follow. The debt accumulated by this industry is by no mean sustainable with such low commodity prices.


Currently (Apr 2017)  the Fed has a policy of reinvesting the proceeds of maturing securities in new securities, so the first option is true.  However, the Fed is believed to be considering ending this reinvestment policy in the next few months.  In that case, the second option will apply- the Fed's balance sheet will decrease and the overall money supply will decrease (assuming that the Treasury will then issue securities to keep its cash balance unchanged).   Option 3 does not really apply because the maturity of a security held by the Fed is not 'revenue'.  They would just reduce assets (Treasury bond) and l moiabilities (cash balance owed to the Treasury) at the same time.


How to get hang of mathematical language that is so extensively used in economics?

If you don't have any background in Real Analysis, I suggest you to read "Analysis I" written by Terence Tao.
I was in a similar situation that you a year ago. I studied Economics at college but we never had to do rigorous proofs (as a matter of fact, I did not know what was a theorem, a mathematical proof, or mathematical logic until I read the book). We only learned some "rules" of derivation and integration, and to manipulate algebraic equations in order to get some important results in Economics. However, a professor suggested me to apply to a PhD program and to take a real analysis course. It was really hard because I had never seen something like that. The books recommended by the analysis professor were the common books you will be recommended here: Baby Rudin, Kolmogorov and Fomin, Ok, Abbot, Apostol, etc. I read some of them (specially Baby Rudin) and I found them too difficult.

The problem was that I found difficult "thinking as a mathematician". Therefore, I started to search for the "best book" in real analysis, and I found this one written by Terence Tao. It was amazing, not only because the deep understanding and the outstanding expository skills of the author, but because he started from the very beginning: he answers some important questions (what is analysis and why to do it?) and starts explaining what are the natural, integer, rational, and real numbers (and also the Peano axioms, and some operations). He also starts explaining what sets are.
Then, he develops the most important concepts of analysis: sequences, limits, convergence, series, etc.

I think the most important feature of the book is that it is completely focused on building mathematical skills: how to prove a proposition, different ways to do it, how to use your intuition to develop a proof, etc. And also, it gives you a complete understanding of the topic. This makes it starkly different from other books such as Baby Rudin (which assumes that you know lots of results that, in fact, you don't), which is useful as a reference book and also when you have some well understanding of the topic (I moved easily from Tao to Rudin when the analysis course I took required me to do so). Another important characteristic of  the book is its appendix: it gently introduces you to the basics of mathematical logic: what mathematical statements are, the structure of proofs, (nested) quantifiers, etc; and gives several examples of proofs.

I often see questions like this and I have always found puzzling why no one recommends this book. It is maybe the best mathematics book I have ever read (as it may seem obvious, I love that book I have fond memories of me learning how to make proofs and what real numbers and analysis really are). Also, it is strange because Terence Tao is unarguably the most important mathematician alive and a very kind person. You can see his blog and the books he has published in the following link: https://terrytao.wordpress.com/books/. I hope this recommendation would be useful.


In most New Keynesian models, can monetary shock affect technology shock also?

No it usually does not.

There is an intuitive reason why: We interpret the technology shock as a residual, the unexplainable part in TFP changes. If we knew that there was a specific causal relationship between monetary policy and future TFP, that part of future TFP would now was explained.

It's true that the MP shock is excluded in standard Solow Regressions, but if you believe there is a causal effect, you need to regress it out of it. Since after all, we understand $A_t$ to be independent.

Note that still, implicitly, through its effects on capital accumulation (which does not exist in standard NK models), monetary policy should impact future TFP. Then, you could make $A_t$ dependent and argue it being a shortcut in the absence of capital.


Are risk-costs a form of external costs?

In the particular case of nuclear power plants, yes, the risk cost can be considered as a negative externality.

That's because nuclear power plant operators are explicitly limited by law in their liability for third-party costs in the event of damages to those third parties. The third-party risk they present could run into trillions (\$/€): at Fukushima, the current cost is estimated at \$200bn and rising. But operators' liability is capped at a much much smaller amount: this differs by country, but typically of the order of a few hundred million to a few billion (\$/€). This means that operators do not have to spend as much money as they otherwise would, on reducing those third-party risks.

If nuclear plant operators had to buy insurance to cover that entire risk, then it's they would be even more uneconomical than they already are. That is, their cost base would be much higher. The direct third-party damages are not paid by the operator, nor by the consumer; but instead are paid by the taxpayers, in the countries that have to clean up after accidents. So, the Japanese taxpayer has borne a large share of the costs, because Fukushima; previously, the USSR did, because Chernobyl - but in that case, some costs were spread all over the Northern hemisphere, due to widespread distribution of the radioactive material; for example, there were restrictions on farming in polluted areas, vast distances from the plant, for decades afterwards.

So, the consequences of the risk are borne as public subsidy. The risk premium itself is not a cost borne directly by anyone: it's an uninsured risk. So, it could be viewed as a negative externality that, if priced in, would move the supply curve.

Note that if you were to refer to these risks, and their consequences, explicitly as subsidies and negative externalities, then you're probably going to have to provide an accompanying explanation (as I've attempted to do above - hopefully you can be more succinct!), as for much of your audience, it won't be immediately obvious as to why you're using those labels.


how could bad harvest season make the Central bank to increase in key rate?

At the beginning of 1800s Britain still roughly $36\%$ of a employment and $33\%$ of British GDP was generated by agriculture (see Grigg 1992). Hence shock to agricultural output could easily lead to recession and have widespread economic effect.
Given what large role agriculture played in the Britain's economy I do not think it is surprising that central bank decided to act in some way. Hiking interest rate in such situation might seem strange with hindsight and living in current fractional reserve system, but that was part of what Keynes called the 'rules of the game'.  As explained by the oxford reference:

Under the ‘rules of the game’, countries losing gold were supposed to raise their interest rates and cut their money supply; countries gaining gold were supposed to cut interest rates and increase their money supply. These rules were intended to restore equilibrium in the balance of payments fairly quickly.

Fall in output (in this case agricultural output) would cause gold outflow because suddenly UK would loose production that was meant to be exported which would worsen its balance of trade. Moreover, under gold standard imports create gold outflow and exports gold inflow. So to play according to the 'rules of the game' the central bank had to hike interest rate.
Of course, doing this in our present fractional reserve system would make no sense and hence it feels very counterintuitive. However, under gold standard central bank cannot just let all gold flow out of country as that would lead to balance of payment imbalances as with gold standard all countries are implicitly under the same fixed exchange rate regime. This was also one of the reasons why eventually gold standard broke down and generally it is not very stable and resilient monetary regime (Eichengreen & Temin, 2010).

Also, how the increase in the key rate could could have made a contribution to the declining prices?

Higher interest rate leads to decrease in prices because it incentivizes people to spend less and save more putting less pressure on prices.


Framing Effect Risk-Aversion Risk-Pursuit

The following is a slight rewording of an example in Richard Thaler's Misbehaving (1):

  Positive framing of certain outcome

    Imagine you are $300 richer than you are today.  You are given a choice between:

      A. A certain gain of \$100.

      B. A 50% chance to gain \$200 and a 50% chance of losing \$0.

  Negative framing of certain outcome

    Imagine you are $500 richer than you are today.  You are given a choice between:

      A. A certain loss of \$100.

      B. A 50% chance of losing \$200 and a 50% chance of losing \$0.

It can be seen that, under either framing, choice A implies \$400 richer than today while choice B implies a 50% chance of \$500 richer and a 50% chance of \$300 richer.  According to Thaler, however, 72% of subjects chose A when presented with the positive framing and only 36% chose it when presented with the negative framing.

To modify this to fit the terms of your question, the initial figures in both framings could be reduced by \$300. The second would then become "Imagine you are \$200 richer ...".  The first would no longer require any imagining.  My (untested) intuition, however, is that the inclusion of imagining in one framing but not in the other might influence people's responses in some way that Thaler's formulation avoids.

I therefore suggest instead the following, because it retains the symmetry of imagining in both framings:

  Positive framing of certain outcome

    Imagine you are \$50 richer than you are today.  You are given a choice between:

      A. A certain gain of \$50.

      B. A 50% chance to gain \$150 and a 50% chance to lose \$50.

  Negative framing of certain outcome

    Imagine you are \$150 richer than you are today.  You are given a choice between:

      A. A certain loss of \$50.

      B. A 50% chance to gain \$50 and a 50% chance to lose \$150.

Reference

(1) Thaler R (2015) Misbehaving: the Making of Behavioural Economics, republished by Penguin Random House UK in 2016, p 33.


Is Adam Smith's "stock" an archaic version of "capital"?

Сельскохозяйственные продукты могут, в виде правила, поступать на
рынок только в таком количестве, чтобы обычная цена их была достаточна
для возмещения капитала, необходимого для доставления их туда, и для
оплаты обычной прибыли.

This is the same place, but from Russian translation of the book. Here "stock" was translated as "капитал", which is a Russian word for "capital".
So, I think yes. It seems that you're right.


The principles of demand and supply are always operating in the background so they are not pointless.

The magnitude of the effect that increase in price has on increase in supply is given by the price-elasticity of supply for housing. If the supply is inelastic then it takes large increases in prices to increase the quantity supplied. In a an extreme case when supply is perfectly inelastic it would not respond to price at all (but this is completely consistent with supply and demand theory - so it would not make it 'pointless', you just always have to take into account the parameters of supply and demand before trying to predict what will happen).
Empirical research (Green, Malpezzi & Mayo 2005) actually shows that housing supply is in many metropolitan areas inelastic so that is one explanation.

In many places government puts heavy restrictions on building of new housing. Especially in historic cities (e.g. Amsterdam) it is excruciatingly difficult to actually build a new housing. Even outside historical cities many places will be heavily zoned. If government restricts supply then of course it will not be able to increase. Also research shows that this is one of the leading causes behind high housing prices (see this empirical study by Glaeser & Gyourko (2002) that examines the effect of strict zoning on prices in Bay Area in US).

You should also not confuse movements along demand or supply curve and changes in demand or supply. Changes in prices produce movement along demand curve. If demand curve is fixed then increase in price (excluding extreme cases such as perfectly inelastic demand or Giffen good) will lead to lower quantity demanded. However, no matter what price is demand can always shift to the right (for example due to change in preferences) which will make people demand more at any price.
If there would be no changes in demand or supply the price should remain constant. The high prices are likely due to the fact that demand for housing is increasing as economies grow and people are having on average higher incomes (since owning a house is likely normal good), and the prices increase because this demand outpaces supply. Of course, that is just gross generalization every city/area will be different. It is not even universally true that house prices increasing in every single area even though on average they do there are areas where they fall (see IMF global housing watch).


What does "endogeneity of the industry structure" means?

Broadly speaking endogeneity means that something some variable is determined within model as opposed to outside it. More narrowly in econometrics it means correlated with the error term (Wooldridge, (2009). Introductory Econometrics: 4ed. p. 88). These meanings are related since one will often lead to the another.
In this paper they actually use it in both meanings.

From economic literature we know that industry structures (e.g. whether industry is monopoly, duopoly, oligopoly, monopolistic etc.) depends on institutional setting and laws as well  (e.g. see discussions of this in Belleflmame & Peitz Industrial Organization Part IV). Moreover, it is often further argued that in turn laws that country implements also depend on "economic conditions of a firm’s country" (Dong, 2019).  So what they mean is that industry structure will not be just given by some exogenous parameters such as let's say availability of local natural resources (this could be an exogenous force that could determine if industry is a monopoly e.g. if there is let's say only 1 viable diamond mine then this would be exogenous factor leading determining market structure to be a monopoly). Rather, here the market structure is shaped by the leniency programs that affect cost of collusion and in turn the leniency programs likely depend on economic conditions (e.g. a country with less problem with cartels might have less anti-trust laws).
Dong (2019) throughout the chapter in previous paragraphs Dong repeatedly states that they are dealing with endogeneity because it could be a valid critique of identification strategy (i.e. their empirical model). Consequently, they also mean it in more narrow way that it would mean that in their regression of margins on those leniency laws the leniency laws would be correlated with error term. In fact, this is why Dong says that they dealt with it by using leniency index from 1990s - using past ('lagged') values instead of contemporaneous one's is a common econometric technique to deal with endogeneity in an empirical model.


Applications of Trig functions in Economics?

The main property of trig functions is their cyclicality. Then one would think that they could be ideal in time series analysis, to model "fluctuations around a trend". I believe that the reasons they are not actually used in such a setting are

1) They are deterministic functions, so they do not allow for fluctuations to be stochastic

2) If the researcher wants to create a model that produces up and down fluctuations (oscillations) around a trend, he would want to obtain that property from the behavioral and other assumptions of the model. If he were to use a trig function, he would a priori impose on the model the sought theoretical outcome.

Instead, one opts for difference-differential equations. There we obtain oscillations (damped or not) if some characteristic roots are complex -and then the trig functions appear, but as an alternative representation, not as buidling blocks.


How to find (document) the actual costs of a specific US government social program?

http://www.fns.usda.gov/pd/supplemental-nutrition-assistance-program-snap - Participation and Costs is the first file in the list

Under the current rules, one problem with stopping activity reports is that all of the statistics you see there would be gone. Another more serious problem is that SNAP recipients are restricted from purchasing certain goods; it's not just that the SNAP card is only accepted at certain merchants.


Why is the supply curve in the foreign-currency exchange market vertical?

Because the supply of pounds in the model is controlled and fixed by the central bank. The central bank does not have to change (increase) the supply of pounds when the exchange rate changes. So the supply of pounds will be just a flat vertical line because no matter what the exchange rate is the supply of pounds will not respond to exchange rate as it is given by the central bank's monetary policy.


Why will inflation raise sellers' purchasing power?

One price relative to others
The key parts of the statement "When a price rises, buyers of that good must pay more, but sellers get more revenue when they sell it. The loss in buyers' purchasing power is matched by the rise in sellers' purchasing power." are "a price" and "that good".
If the price of oranges increases from \$1 to \$2, then it does not imply that 2 dollars post-inflation is worth the same as 1 dollar before the price rise; we could expect some devaluation, but it would be proportional to the impact of oranges in the whole economy.
Oranges increasing in price means that the purchasing power of people who buy oranges is decreased, but the purchasing power of people who sell oranges is increased, since they can buy more stuff (e.g. apples) from their orange-derived revenue than before.


The Difference between investment and savings in CA identity

Yesterday I gave an example of an economy based on grains of wheat where the real domestic activities were

Plant wheat grains, grow them and harvest them, getting more grains as a result
Consume wheat as food
Store unconsumed wheat for future planting or consumption

In an open economy there are two further possibilities

Send wheat grains abroad (e.g. exports)
Receive wheat grains from abroad (e.g. imports)

So saving represents the excess of production of wheat grains over consumption, while investment represents a combination of planting grains for future production and increasing stocks of grain.

In this open economy, you then get the macroeconomic tautology that the increase in stocks must be equal to production plus imports minus the sum of consumption and use for planting and exports.  Rearrange this and you get that saving (production minus use for consumption) minus investment (use for planting plus increase in stocks) must equal exports minus imports (in effect the balance of trade or, if you take into account other foreign income, the current account balance), i.e. $S-I = X-M$ or $CA=S-I$

You may then want to translate this into monetary terms using the price of wheat, or in a more complicated economy the price of all the stuff involved, but the key is to recognise this is about stuff measured in monetary terms rather than about money


What is the empirical price elasticity of demand for insulin?

You are right! Textbooks are probably not careful enough.
However, there is evidence that the most inelastic drugs are those indicated for the treatment of chronic diseases with price-elasticity ranging form -0.08 to -0.03 [see here], while studies establish a general inelastic price-elasticity of drugs of around -0.2 (here). In the same Plos One paper, therapeutic groups are sorted by price-elasticity and the diabetes group is ranked third behind cardiovascular and endocrine diseases (see Figure 1).


How to formulate that a factor may influence a variable or the changes in that variable?

"may also influence $\Delta I$" seems better to me. Of course for all $s \geq t$ the value of $I_s$ is also influenced by $A_t, B_t, C_t$, but this is an indirect effect. One could say it is fully encoded in $I_{t+1}$.


Government Spending vs. Investment Spending

Government spending is the spending by the government on goods and services (non-capital goods). This means that the government buying physical capital would not be included in government spending, and would be included in investment spending.


Taken right from the society for computational economics:

  Computational economics explores the intersection of economics and
  computation. These areas include agent-based computational modeling,
  computational econometrics and statistics, computational finance,
  computational modeling of dynamic macroeconomic systems, computational
  tools for the design of automated Internet markets, programming tools
  specifically designed for computational economics, and pedagogical
  tools for the teaching of computational economics. Some of these areas
  are unique to computational economics, while others extend traditional
  areas of economics to new areas through computational techniques.

I would say that calculating cost of manufacture is too trivial to fall under this category - unless it's a monopolist who has to take into account equilibrium responses of his suppliers and demanders to his choices.


Why does Slutsky compensation "overcompensate" the consumer?

Here's a figure to explain:

Starting from the old price line, where the optimal consumption bundle is point $A$, we increase the price of $y$ to get the new price line.

The Slutsky compensation says that we have to give the consumer enough extra income so that he can afford to old bundle ($A$) at the new price. Thus, we shift the new budget constraint out to the dashed line.

The reason this "overcompensates" is that, taking the dashed line as the budget constraint, the consumer could afford bundle $B$, which gives him higher utility than he started with.


Is it the most efficient to allow free transactions in financial markets?

As a part of the overall financial system, such instruments can increase market efficiency and facilitate exchanges.  Since all such instruments are essentially backed by government monopoly rights (at the most basic, the monopoly held by governments to create money) they would fall under "land" in terms of the classical factors of production.  While it may seem strange to count financial instruments as a form of "land" it makes sense if you understand "land" to really mean "any factor of production associated with monopoly / exclusive use rights."  As a form of land, the returns to financial instruments are thus economic rent and the root cause of speculation.  So yes, they should be regulated / taxed.

Note, however, that it is not true that they "create no value economically" -- monopoly rights exist for a reason, often specifically to increase overall market efficiency.  It makes more sense for government to grant rights-of-way to utilities and such, to prevent redundant over-building of utility lines, etc.  But the value so created is being created by government, not the private entities that often claim the rent as their own.  Hence they should be taxed at (nearly) 100% so that the rents go back to the community, rather than free-loading rentiers.

There is plenty of theory on all of this, though it is considered heterodox today.  Henry George (and his peers) wrote about it at length around the turn of the 19th-20th century, and economists such as Mason Gaffney continue to publish current works in this area today.  The New Physiocrats are another contemporary group publishing ideas along the same lines.


Understanding Price Elasticity of Supply

To elaborate on 123's comment, crude oil is extracted from the ground and can be manufactured/refined into a variety of produces (vehicle fuel/plastics/kerosene/etc.) If the price of kerosene increases, then it is natural for produces to want to substitute from producing plastic to producing more kerosene. Even if the supply of oil (of all varieties) is inelastic, the supply of individual oil-based products need not be.


Value added by sector of economic activity

The first line of your World Bank link gives you the explanation:

  Industry corresponds to ISIC divisions 10-45 and includes
  manufacturing (ISIC divisions 15-37)

Hence, if you add up industry and manufacturing you count some activities twice, the divisions 15-37 that is.

But even if you take this into account you should not find 100% if you are adding up the value added of all the activities. Gross domestic product (GDP) is not the sum of the gross value added (GVA) of all the activities but:

  GDP = GVA + Taxes on products - Subsidies on products

According to Statistics Explained

  Gross value added of the total economy usually accounts for more than
  90 % of GDP

So, the message is to check carefully the definitions and concepts used by the World Bank.


Are homothetic preferences strictly monotonically increasing?

Homotheticity requires that

$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$

This is not defined over the "increasing" part of strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds.

Are nomothetic preferences weakly monotonic?

However, the "strict" part (in both jointly) is required, i.e. $U(\alpha x, \alpha y) \neq U(x,y)$ for $\alpha \neq 0$.. You cannot have homothetic and weakly increasing or weakly decreasing preferences.

Proof outline To prove that, we assume otherwise. There is some homothetic $U$ with a strictly increasing part and a flat part. You will see that the $\gamma$ corresponding to the increasing part can potentially be any number (it depends on the functional form). However, the $\gamma$ corresponding to the flat part needs to be 0.[1] Hence, there cannot be a $\gamma$ that holds for the utility function as a whole, for any general increasing part.

[1]: Unless of course, $U(x,y) = 0$ at the flat part. However, as utility functions are immune to monotone transformations, we can ignore such cases.


The true answer is of course it depends. What it depends on is how you define sustainability.

In a natural resources context, with weak sustainability, soil quality is a capital asset, just as a fish stock, a forest or an oil field. Given that we have manure and fertilizer I would argue that it is a renewable resource (until we run out of phosphate perhaps or poison the soil with too many pesticides), but let's start from the simpler case of a non-renewable resource.

In that case the golden rule tells us to gradually exhaust the soil until no more degradation is possible. In the simplest case we should degrade at the rate of interest. You correctly observed that in principle there are no externalities. The individual optimal solution is equal to the social optimal solution, provided the individual discount rate (interest rate) is equal to the social discount rate (see e.g. Clark 2010 for this often missed point).

One can modify such a model to allow for renewable resources with the use of fertilizer or natural regeneration, which would change the solution somewhat. Kenneth McConnel (1983) has a model showing the basics.

The reason that soil exhaustion is optimal is that the proceeds can be invested in alternative assets, which generate a larger rent. The problem is of course that i) this model assumes we discount the future (which we do) so future generations are discounted; ii) such models at an economy wide level typically assume the existence of a backstop technology, something we can use when we run out of good quality soil. Whether or not something like that will exist is debatable.

Moving towards a strong sustainability viewpoint the main thought is that we should not deplete our natural resources beyond their regenerative capacity, unless we already have a substitute. We don't currently have a real good substitute for soil quality, although we can use fertilizers and such to make up for some of that. From a strong sustainable perspective we are therefore not (yet) sustainable.

From a bit broader perspective the production of artificial fertilizer is very energy intensive, and one can question whether the current amounts used (as well as the earlier mentioned pesticides) do not have unintended effects, but then you're back at externalities.


Structural Modelling of Expectations/Beliefs

Check for example Postel-Vinay and Robin (2002) and references therein to see how people go about structurally modeling the labour market.

The general framework is a dynamic one, where the job-search process is a Poisson arrival process for each worker, and the results are derived by finding the market equilibrium conditions.

The interesting point in this literature is that indeed labour market decisions are based on expectations about the employers, but here specifically the expectation about how career trajectories diverge and what this means for reservation wages for the employee, and how this depends on the productivity of firms.
In the cited paper, the productivity determines what a firm may be able to pay eventually. It raises wages when competing offers arrive by firms that may theoretically also employ the worker.

The worker forms expectations about his future trajectory and this implies that he is willing to work for a high productivity firm for less money, simply because he expects it can raise wages in the future if competing offers arrive.

So you see that expectations are indeed important for the job-search process. For your idea, you have to be specific as to what the worker expects if a firm is racist. An interesting idea would be to think about blacklists and stigma, which would mean that the worker could be locked-in the current firm. This would require the firm to offer higher wages compared to others.
It's a straightforward mechanism, and interesting I think, but you need to show basically how this impacts the workers choices based on preference (since you are doing structural modeling). It may be a good idea to start with reduced form analysis, and also to check out the sociology literature as I am sure this sort of idea has been used before. Your contribution would be to model it and get exact parameter values, run counterfactuals and hopefully discover some surprising results.

What worries me a bit about the idea is that you are somewhat missing an objective for the firm to be racist, given that it optimizes everything else in equilibrium in this setup. Of course you may assume it the same way one assumed productivity, and while it may be econometrically sound, it is not really identified in a theoretical sense. What I mean is that racism would just be a parameter akin to pushing down what a worker expects from a firm in terms of future productivity. And since it is outside of the explanatory power of a structural model, it may be that structural modeling is not really the necessary approach - a reduced form will tell you as much.
Perhaps you can make it bite, by making the stigma structurally different from lower productivity, and perhaps exactly this matters. This would be where the result would be interesting to me.


If an item is rarer than others, is it that much more valuable?

(1) What value an item has to you depends on your preferences, not scarcity.

(2) The market clearing price of an item will depend on the preferences of market participants and the supply schedule ("scarcity") of the item.

The way you seem to be thinking about value is very "non-economic", if that's a word.


How did downgrading bonds contribute to Greek crisis?

Here's a non-technical answer:

Bonds are a form of debt.  What the issuer is selling is essentially a promise to repay the principal (i.e. whatever price the buyer paid) and some interest to the buyer.  (Note: since we're talking about debt, you could also think of the buyer as a lender.)  In return, the issuer gets to use the money from the sale for whatever current projects needed funding in the first place.

Skirting some technical stuff, assume the yield is the same thing as the interest rate.  When lenders think there's a higher risk of not getting their money back, they'll either 1) not lend in the first place, or 2) if they do lend, demand higher a higher interest rate to compensate them for the higher probability of not getting repaid.

When someone downgrades your credit, it's a statement about you: they believe you're at a higher risk of not fulfilling your promise to repay the lender.  So any lender that does decide to take a risk and loan you money (i.e. buy your bond) is going to be in situation #2 above: they'll demand you pay more interest on the loan (here, higher bond yield) to compensate them for your riskiness.  That makes borrowing more expensive for you since you now have to pay a higher interest rate than you would have if you had better credit.

Rollovers are a type of reinvestment.

At the heart of the matter, credit downgrades happen when lenders lose faith in your ability to fulfill your promises.  This happens for a multitude of reasons.

As an example, S&P's 2011 downgrade of U.S. debt happened because they believed 1) that the pattern of running deficits most years might eventually put the U.S. at risk of not repaying (that is, the debt-to-GDP ratio might not be sustainable), and 2) that political leaders lacked a credible plan to tackle the problem.


Hint:
The $T$ is variable itself so the total differential should read as:
$$dY = C'dY - C'dT + I'dr + dG \implies dY = \frac{1}{1-C'}I 'dr + \frac{1}{1-C'}dG - \frac{C'}{1-C'}dT$$
Furthermore, by Fisher equation $r=i-\pi$. Hence we have:
$$dY = \frac{1}{1-C'}I 'di - \frac{1}{1-C'}I 'd\pi + \frac{1}{1-C'}dG - \frac{C'}{1-C'}dT$$
$$(1/P)dM - (M/P^2) dP = L_r'dr + L_Y' dY - L_T'dT$$
again use the Fisher so:
$$(1/P)dM - (M/P^2) dP = L_i'di - L_{\pi}'d \pi + L' dY - L'dT$$
This is IS-LM model so you are looking for equilibrium output and nominal interest rate $dY$ and $di$. You have two equations and two unknowns. You can solve it even with substitution but you should probably follow the textbook's advice to use Cramer's rule.


Lack of skilled IT workforce, but they do not raise wages

The factual observations you've listed fit neatly with the law of supply.  You've seen that wages (price) have fallen, and during the same period, supply has gone down.

The fact that companies publicly complain that there aren't lots of cheap, highly skilled workers available is not the same as companies being willing to convert that into effective demand by raising wages.

One common reason companies moan about things publicly, for instance, is that they want the government to intervene to help them.  In this case, perhaps they want government to provide more subsidised IT training, or permit more immigration in the sector, either of which could lower wages further or prevent them from rising.


CES utility function in an Edgeworth box

There seems to be some confusion in the expression for $x^*_i$ in the question that whether $i$ is for consumer of for the good. Assuming $i$ is for consumer:
Let $x^*_i = (x_1^i,x_2^i)'$ be the equilibrium bundle for consumer $i$.
Since utility function is same for both, from MRS we have:
\begin{align}
\frac{x_1^i}{x_2^i}=\bigg(\frac{p_1}{p_2}\bigg)^{s-1} \tag{$i=1,2$}
\end{align}
Budget constraint for $i$:
\begin{align}
p_1x_1^i+p_2x_2^i&= p_iw  \\
x_2^ip_2 \Bigg(\bigg(\frac{p_1}{p_2}\bigg)^s+1\Bigg)&=p_iw \tag{using MRS}\\
x_2^i \bigg(\frac{p_1^s+p_2^s}{p_2^{s-1}}\bigg)^s&=p_iw \\
x_2^i &=\frac{p_2^{s-1}}{p_1^s+p_2^s}p_iw
\end{align}
So,:
$$x^*_i(p,pw) = \Bigg(\frac{p_1^{s-1}}{p_1^s+p_2^s}p_iw,\frac{p_2^{s-1}}{p_1^s+p_2^s}p_iw \Bigg)$$
The question can be solved further, for $p_1/p_2$ using the constraint: $x_j^1+x_j^2 = 1$


Form of security with a potentially infinite amount of equity for a continous payout for game?

One "natural" approach would be to base the price on the expected discounted revenue stream. Writing $R_t$ as the (random) revenue in period $t$ of one share, this then would be:
$$
p_t = \mathbb{E}\left(\sum_{t = 0}^T \frac{R_t}{(1+r)^t}\right) = \sum_{t = 0}^T \frac{\mathbb{E}(R_t)}{(1+r)^t}.
$$
In addition, you might want to lower this price a bit, especially if the income stream is very volatile and if you would expect that individuals are risk averse.
One thing determine then is the rate $r$. Ideally, this should be set equal to the rate of return of a risk-free asset (if you have this in your game). If the game does not last very long, I guess setting $r = 0$ also makes sense.


If you mean you want to convert ECU to EURO, they are essentially the same currency; one was a basket of European currencies used before 1 Jan 1999, the other is used after. The exchange rate is exactly 1 ECU = 1 EURO (reference) So you should be able to use that as EUROs.

Note: This came from my comment above, which I take that it helped the OP. Just to boost up answered questions ratio for this site.


Hedonic regression and attribute-shares in house prices

What you are proposing is an unusual application of hedonic regression in respect of house prices. The method is most commonly used for one of two purposes:

a) To explain and perhaps predict relative house prices.

b) To estimate the value of characteristics of houses and their neighbourhoods, especially those which are hard to value in other ways such as non-market environmental goods.

When the method is used for either of these purposes, negative parameters are unlikely to be viewed as a problem.  Taking one recent study as an illustration, Mora-Garcia et al (2019) estimate hedonic regressions with negative parameters on various regressors (see Table 6, pp 15ff).  Their discussion (p 25) notes the negative parameters without concern (focusing rather on the consistency of the negative signs between different models).   When the aim of a hedonic study is to estimate the effect on house prices of an evironmental ‘bad’ such as air pollution, a negative parameter on the pollution regressor is to be expected.  Komarova (2009), for example, found negative parameters for several air pollutants in Moscow (see Table III p 323 and Interpretation p 324).

It’s only your particular aim of analysing house prices into positive cost shares that makes negative parameters a problem.  There isn’t a standard way of dealing with the problem because for most studies it isn’t a problem.

There are however some ideas in the literature that may be helpful.  In the case of distance to work (and generally of distance to any amenity), an alternative is to use some measure of accessibility with the property of associating greater ease of access with a larger regressor value.    Heyman et al (2019) review various measures of accessibility used in hedonic studies.  In particular their Equation (3) in section 3.3.2 contains a term of the form $e^{-c}$, where $c$ is cost of travel between zones, giving that measure the desired property.  Franklin et al  (2003) present a rather sophisticated index of accessibility to work, also with the desired property, that takes account of multiple work locations with different numbers of employees (see  Equation 2 p 5).

A general approach that could be used for any regressor that is found to have a negative parameter would be to derive from it a measure defined so as to vary in the opposite direction.  Suppose for example that age of property $A_i$ was found to be significant and to have a negative parameter, and the highest age in the sample was 100 years.  One could simply replace it by a regressor defined as $100-A_i$.  Or one could define a scale of discrete points, say 5 for 0-10 years, 4 for 10-25 years, and so on.  Trying out different measures of this kind might have the additional benefit of improving the fit of the regression.


It is just a strict preference relation, an asymmetric (generic) relation $\succ$ is such that $a \succ b \implies \lnot b \succ a$.
With respect to symmetric preferences there are not really  pure symmetric ones in consumer theory at least, the only symmetric relation is total indifference ~ over the whole choice set. Clearly a ~b implies b ~a. A typical $\succeq$ preference relation is the union of an asymmetric and symmetric part.


indifference curve slope from utility function

Hint: First find the total derivative of $u(x_1,x_2)$, set it to zero as utility does not change along an indifference curve, then solve for $dx_2/dx_1$


Minimum sample size for cross-sectional RDD

I've run RDDs on similar sample sizes and got the paper accepted at a respectable conference. It's important to run McCrary Density Tests on your sample to look for people using knowledge on the treatment assignment rule to manipulate results and - even more important - to have a clear identification strategy, a qualitative justification for what you're doing.


Is a mixed strategy ever the best response to a pure strategy?

Fixing the strategy of the opponent, a mixed strategy never yields a strictly higher utility if you are expected utility-maximizing.
The reason is that the expected utility from a mixed strategy is at most as high as the highest utility from the pure strategies which this mixed strategy plays with positive probability.
That is not to say that a mixed strategy cannot be a best response, though.
Consider for example rock-paper-scissors and a imagine your opponent playing each of the three actions with equal probability.
This leaves you indifferent between all three actions and hence any mixed strategy is a best response.

Things can be different when players are not expected-utility maximizers. If that's what you're interested in, you could take a look at the paper "Equilibrium without independence" by V. Crawford, 1990, as well as some of the references therein.


One is Relative Utilitarianism (RU). Under the axioms below, society's preference can be represented by the simple sum of individuals' vNM utilities (each normalized to between $0$ and $1$). That is, every individual is given equal weight.

Pareto Axiom. (This is just the usual: If everyone prefers lottery $p$ to $q$, then so too does society. And if everyone strictly prefers lottery $p$ to $q$, then so too does society.)
Separability.
Invariance.
Anonymity. (Again, this is just the usual, though a small technical adjustment may be necessary.)

The original papers on RU are Dhillon (1998) and Dhillon & Mertens (1999). However, the first contains fatal mistakes and the second is indecipherable. See Börgers & Choo (2017) for a simple and correct exposition/proof of RU.

(Of course, the literature on utilitarianism and what you ask is vast. This is just one small example.)