Can intermediate good producer's profit ever be zero in new keynesian model? In monopolistic competition, firms are said to have zero long-run economic profit.

This makes me wonder whether profits of intermediate goods producers in New Keynesian model ever be zero.

For reference we can assume Gali's simple New Keynesian model.

Yes. In the core New Keynesian model in Galí, it is possible for the aggregate profits of intermediate good producers to be zero or negative; and it is even more likely that the profits of individual intermediate good producers, who may be stuck at prices far from the optimum, will be negative.

Since it's easier to characterize the aggregate, I'll discuss the conditions there. Note that the aggregate profits of intermediate good producers are just aggregate revenues minus the aggregate wage bill, or
$$\Pi =PY-WN$$
where $P$ is the aggregate price index, $Y$ is aggregate output, $W$ is the nominal wage (common to all producers thanks to the competitive labor market) and $N$ is aggregate labor input.

Now, consider the special case where intermediate producers have constant returns to scale and productivity is normalized to 1. There we have, to first order, $Y\approx N$ (there is a second order difference due to price dispersion). Hence, again to first order, we have $$\Pi \leq 0 \Longleftrightarrow P\leq W$$
When is $P\leq W$? Well, using lowercase variables to denote logs, we know from household optimization that $w - p = \sigma c + \varphi n$ (see Gali p.43 equation (2), for instance), where $\sigma$ is the inverse EIS and $\varphi$ is the inverse Frisch elasticity. And since market clearing and our simple production function imply that $y=c=n$ to first order, this becomes $$w-p = (\sigma + \varphi)y$$
Finally, we observe that at the "natural level" of output $y^n$, markups should equal their desired level, which is $\mathcal{M}=\varepsilon/(\varepsilon-1)$, where $\varepsilon$ is the elasticity of firm demand. Writing $\mu =\log \mathcal{M}$ and substituting into the equation for $w-p$, this implies $-\mu=(\sigma+\varphi)y^n$. Finally, subtracting from the equation for $w-p$, we have
$$w-p = (\sigma+\varphi)(y-y^n) - \mu$$
We'll have negative aggregate profits $P\leq W \Longleftrightarrow w-p \geq 0$ when $(\sigma+\varphi)(y-y^n) \geq \mu$. Note that this happens when output is high, since profits in the basic NK model are countercyclical.

Interestingly, under some not-totally-crazy calibrations this can happen. For instance, if we assume that both the EIS and Frisch are 1/2, so that $\sigma=\varphi=2$, and also assume $\varepsilon = 10$, so that $\mu = 0.105$, then exceeding the natural level of output by even 3% is enough to deliver negative aggregate profits, since $4\times 0.03 > 0.105$. And again, note that it is much more likely that we'll have individual firms with negative profits; in the case where aggregate profits are even close to negative, any firm unlucky enough to be stuck at a much lower-than-average price will have negative profits.

Finally, I should clarify that all of this is unrelated to the concept of zero long-term economic profit that you mention. This concept is often embedded in models of monopolistic competition in the form of a free-entry condition (where firms pay some fixed cost ex ante to exist), but this is very rare in New Keynesian models - presumably because NK models focus on business-cycle frequencies where the entry margin is limited.


What's a "correct price" really? Does it exist? Lately I've been thinking about the concept of price setting in market economy. There are various (perhaps naive) view on this such as:

The price is always the optimal for the buyer and the seller.

or

The prices reflect collective valuation of a particular resource/service/...

However, I find these unsatisfying due to the complexity of the concept of "value". Particularly how one's own choices relate to choices of others. Then, what's a "correct price" really? Is it "the price I paid" or some "communal price", "societal price"?

tl;dr as KennyLJ correctly points out the concept of correct price is meaningless and it does not exist or rather you could say any price is the correct price (but again this just makes the whole concept meaningless because if every price is correct then thats just a price -  a concept of 'correctness' makes only sense if there are some incorrect instances).
The reason for this is that there is no 'intrinsic' objective value to anything. Since all value is completely subjective without even slightest objective component there cannot be any correct or incorrect price because there is no correct or incorrect value to anything, every valuation and any price is correct and hence the word 'correct' loses its meaning.

Full Answer:
The question of what is the 'correct' price is actually interesting from the perspective of history of economic thought and it actually dominated economic thought in ancient and medieval times and in some non-economic thinking it persists even to this day. In those times what is correct was heavily influenced by ethical considerations as at that point economics did not yet emerged from moral philosophy and could not be divorced from all other subject subsumed there (similarly as all natural subjects were at the time part of natural philosophy). As a result the early economic thinking was dominated by the concept of 'just price' which was considered both moral and desirable. Furthermore, just price and any other thinking on 'correct prices' relies on notion of objective intrinsic value.
Paul Mueller & Jan Gerber discuss the history of 'just price' concept in great detail I will do it here only briefly. The concept already originated with Aristotle in his Nicomachean Ethics and Politics. In Aristotelian thinking a just price would be a price at which the same value is exchanged. That is a just price wold be a price at which $\\\$1$ of value embedded in apples exchanges for exactly $\\\$1$ value embedded in peaches.
Thomas Aquinas expands on the Aristotelian concept according to Muller & Gerber by using the following innovations:

While it is still wrong to sell something for more than it is worth, it is not wrong to sell something for more than one paid for it. While Aristotle saw injustice in the shoe example, Aquinas recognized that one can do something productive or useful with a good that improves it in such a way that its value is higher, and so a higher price is still just. Transporting the shoes from one town to another represents a kind of improvement, as does any kind of labor performed on the good.
While this is a little more sophisticated than Aristotle’s account, it still relies on an “intrinsic” or “natural” value of goods—with a focus on the costs or input which creates value. Furthermore, Aquinas and other scholastics were also wary of profit. Some profit, a “reasonable” amount, was fine. But excessive profit could only be the result of injustice and exploitation.

Later when economics started to emerge as a separate field from moral philosophy classical economists started to turn away from the moral notion's surrounded around the 'just price' (since in science we like to make sharp distinction between normative value judgement and positive inquiry) although as the article by Muller & Gerber explains in moral philosophy the notion lived on for a bit more, but the classical economists such as Adam Smith or David Ricardo did not turned away from the notion of intrinsic value which would still enable a possibility for 'the correct' price to exist.
In fact Adam Smith was firmly behind the idea that there is some intrinsic value to goods and services. As Steven Horwitz writes:

Smith is very clear in The Wealth of Nations that he sees labor as the source of value. For example, in the opening paragraph of Chapter 5 on real and nominal price (I.v.1., p. 47), he writes:
The value of any commodity, therefore, to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labor which it enables him to purchase or command. Labor, therefore, is the real measure of the exchangeable value of all commodities.
And later in that chapter (I.v.7., p. 51):
Labor alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price;  money is their nominal price.

However, as our scientific understanding progressed any notion of objective intrinsic value got completely shattered. This occurred during the marginal revolution where subjective theory of value was virtually simultaneously discovered by Jevons, Walras, and  Menger in the late 19th century.
The subjective theory of value claims that there is no objective intrinsic component to any value and all value is completely subjective given by person's own preferences. This theory of value can explain situations that labor theory of value or other theories of value cant. For example, under labor theory of value digging hole and immediately covering it up should have some intrinsic value due to labor component embedded in it but we can observe that people dont value such absurd waste of labor. This can be fully explained by subjective theory of value that would simply postulate that the dig up and immediately covered hole is worthless because nobody values it (or if someone actually pays for it it has some subjective value of them). Conversely, even things that do not have any labor embedded in them can be valuable. Furthermore, value further fluctuates based on what are people's marginal utilities. As the quotes mentioned in the_rainbox's answer from Adam Smith show, classical economists already argued that individuals have different levels of utility or 'value in use', but did not properly connect them to prices, or 'value in exchange', considering them separate, one derived from the quantity of labour input and other production factors (see Stigler's The Development of Utility Theory. I).
Currently subjective theory of value dominates the mainstream economic thinking, and does so to the point that with perhaps a little bit of a hyperbole the detractors from this theory are in similar number and position as anti-vaxxers in medical science. You will find the subjective theory of value in every modern economic textbook and arguably it is either implicit or explicit in wast majority of the whole corpus of economic literature written post marginal revolution. What even more, the subjective theory of value 'spilled' from economics back to the moral philosophy where it was used to nail the last nails into the coffin of the moral theory around 'just price' (see the article by Mueller and Gerber again).
As prefaced at the begging under subjective theory of value, objective intrinsic value that could be used as a justification for correct price simply does not exist. If two people make any voluntary transaction in the market they must either exchange equal or both greater value than was lost (or some combination of thereof) meaning if any transaction occurs the price is automatically the 'correct' price for people engaging in that transaction. Furthermore, even if a transaction does not occur because maybe seller asks for too large price or buyer for too low price you cannot dismiss their subjective valuations as incorrect and the prices they are asking for as an incorrect prices. Once you adopt subjective theory of value any notion of correct price becomes completely absurd. Hence the answer to your question is that there simply isnt any 'correct' price.


Why wouldn't competition prevent "usurious" payday loan rates? The current Wiki page on "Payday Loans" claims loans are priced above marginal cost. The justification is that

  If a lender chooses to innovate and reduce cost to borrowers in order to secure a larger share of the market, the competing lenders will instantly do the same, negating the effect. For this reason, among others, all lenders in the payday marketplace charge at or very near the maximum fees and rates allowed by local law. [23]

Doesn't this logic contradict the fact that competitive markets price goods at their marginal cost?

Is there an implicit assumption that the payday loan market is not sufficiently competitive? If so, is this assumption true?

The posted quote is economic nonsense.

  If a lender chooses to innovate and reduce cost to borrowers in order to secure a larger share of the market, the competing lenders will instantly do the same, negating the effect.

This applies to any industry without intellectual property protection -- it is hardly unique to the payday loan industry. By this logic, we'd expect to encounter massive price gouging across dozens of industries.

Besides, if payday loan innovation is in the form of software that better predicts default (the most likely path), it will be protected under copyright law and potentially software patents. And while business model innovations are not patentable, there's still a first mover advantage.

The payday loan business is not highly profitable.

Profit margins of payday-loan corporations are publicly available, and lower than most other industries. One study found that "despite the common belief, payday lending firms do not always make extraordinary profits. In fact, when compared to many other well-known lending institutions, payday lenders may fall far short in terms of profitability."

This is not surprsing, since the payday loan market is highly saturated, which suggests substantial competition.

"Usurious" APR rates are misleading.

A typical payday loan charges \$17 for a two-week \$100 loan.
Expressed as an annualized rate, this is an "outrageous" 390% APR.
But the loan's short-term nature means transaction costs will likely prevent a large profit.
(The source of this information is potentially biased. Be sure to read critically.)


Pareto optimality and Externalities Let's consider 5 farmers, each of them has 2 cows to put into the field.
So every farmers can put 0,1 or 2 cows. I denote the three stategies by $q_i$, i=0,1,2.
Now, the payoffs ( i.e. the amout of food that each cow will eat) are given by:
$$
u_{i}(q_1,...,q_5)=q_{i}(12-(q_{1}+...+q_{5}))
$$
Now, for finding the Nash equilibrium I draw the matrix which has on the row one player and on the columns the others four. I observe that strategy 2 stricly dominates 1 so the N.E. is given by (2,2,2,2,2) and the correspinding payoffs are (4,4,4,4,4).
(If I estend the strategies to $q_{i}\in[0,2]$ the N.E. remains the same, right?)

Now, the Pareto optimality, if I have understood well should be the strategies (1,1,1,1,1) which give me as payoffs (7,7,7,7,7). Are there other pareto optimality profile? Because I think that I have understood the problem for 2 players but for more I'm not totally sure.

Then, the last question :) If I tax (c) farmers that put two cows in the field, how much I should tax to inforce N.E?
For replying to this I have think in that sense: for strategy I have that my payoffs functions are given by
$$
u_{i}(q_1,...,q_5)=q_{i}(12-(q_{1}+...+q_{5})-c)
$$
Hence, if I look to the payoff matrix in order to have strictly dominance of strategy 2 I should impose c=1.

The idea behind Pareto optimality is efficiency. On the discrete set, there is no allocation where, compared to $(1,1,1,1,1)$, we could make someone better off and keep the others at least indifferent. This is why this allocation is Pareto optimal. However, any other efficient allocation (in the sense that we are not throwing away resources) will be Pareto optimal here as well.

For the set in $\mathcal R$, we assume a symmetric solution and solve

$$\max_{q\in[0,2]} 5\cdot q(12-(5q))$$
$$\Rightarrow 5(12-5q) -25q$$

The first term is the direct benefit of increasing $q$, the second one is the externality (that we now internalize). The interior solution is given by $q = 1.2$ where everyone gets the payoff $7.2$. We know that the interior solution solves the problem given that the boundaries $0, 2$ both delivered lesser value than $q=1$.

Finally, for the tax, derive the FOC of the agents given a general tax $c$. Then, look for the optimal quantity you want them to have, that is, replace $q_i$ with $q_i^*$, the welfare-maximizing solution, and then look for the value of $c$ that would solve the FOC (and hence make the farmers optimally chose what you want them to).


GNI versus GNP for Measuring Development In the calculation of the Human Development Index, Gross National Income (GNI) is used rather than GNP.

I've searched around on the internet, but I haven't found a clear explanation of what the precise difference between GNI and GNP is. Wikipedia defines GNI as "total domestic and foreign output claimed by residents of a country" and the GNP as "market value of all the products and services produced in one year by labour and property supplied by the citizens of a country." Just like the income and expenditure approaches to calculating GDP yielding the same result, it seems to me that GNI and GNP should be at least theoretically equal.

What is the difference between GNP and GNI, and why use one over the other for measuring development?

GNP measures production, so is a better measure of how the country is doing in the long term at accumulating fixed capital and developing. Fixed capital has a much higher return than the financial capital that GNI is typically derived from. However, both measures are usually very similar, except for some tax havens such as Ireland that have significantly more GNI than GNP.

Suppose a country consisted of one individual making trillions of dollars on the stock market and boosting GNI without any national product. If they don't own any fixed capital, this would be a very bizarre economy and not really "developed".

On the other hand if a country has lots of production but has a negative GNI because, say, a domestic company issued a massive dividend to overseas investors that year, then the country could still be called "developed" because it has real people and factories and so forth.

Neither of these cases have anything to do with reality of course, they're just examples.


Difference between yield curve and term structure of interest rate Can somebody please explain me the difference between yield curve and term structure of interest rate?
Thanks

As far as I understand these terms are used almost interchangeably with the former term being more wide while the later applies more specifically to visualization of the term structure.
According to Mishkin and Eakins, Financial Markets and Institutions. 8ed, in chapter 5 pp137 on Term Structure of Interest Rates:

A plot of the yields on bonds with differing terms to maturity but the same risk, liquidity, and tax considerations is called a yield curve, and it describes the term structure of interest rates...

Also the whole chapter on term structure of interest rates focuses almost solely just on the yield curve.
However, as I understand it, the term structure of interest rates can be also applied also more widely, while yield curve specifically refers to the above mentioned plot. However, since the yield curve is almost always used to visualize the relationship the terms are colloquially often used as synonyms.


Is the Convex Combination of Two Pareto Optimal Allocations Also Pareto Optimal? I am studying for my qualifiers, and I ran into this question from a previous year's exam.

$\textbf{Question:}$

Consider a two-consumer two-good pure exchange economy. Both preferences are locally non-satiated and convex. Prove or disprove the following statement: if $(x_1,x_2)$ and $(\hat{x}_1,\hat{x}_2)$ are two different pareto optimal allocations, then the convex combination, $(\alpha x_1+(1-\alpha)\hat{x}_1,\alpha x_2+(1-\alpha)\hat{x}_2$ MUST also be pareto optimal for any $\alpha\in(0,1)$.

I believe the statement is true, and here is the work for my proof below.

$\textbf{My Proof:}$
By pareto optimality of $x_i$ and $\hat{x}_i$:
$$\not\exists\; x_i^\star\; s.t.\; u_i(x_i^\star)\geq u_i(x_i)\;\forall i\; \text{and}\; u_i(x_i^\star)> u_i(x_i)\;\text{for at least one }i \;\text{or}\;u_i(x_i^\star)\geq u_i(\hat{x}_i)\;\forall i\; \text{and}\; u_i(x_i^\star)> u_i(\hat{x}_i)\;\text{for at least one }i$$
$$\implies\;u_i(\alpha x_i)\geq u_i(\alpha x_i^\star)\;\forall i\;\text{and}\;u_i((1-\alpha)\hat{x}_i)\geq u_i((1-\alpha)x_i^\star)\;\forall i$$
$$\implies\not\exists\;x_i^\star\;s.t.\;u_i(\alpha x_i^\star+(1-\alpha)x_i^\star)\geq u_i(\alpha x_i+(1-\alpha)\hat{x}_i)\;\forall i$$
$$\text{and}\;u_i(\alpha x_i^\star+(1-\alpha)x_i^\star)> u_i(\alpha x_i+(1-\alpha)\hat{x}_i)\;\text{for at least one}\;i$$
$$\implies (\alpha x_i+(1-\alpha)\hat{x}_i)\;\text{is pareto optimal}$$
$$\blacksquare$$

This proof seemed almost too easy, so I'm wondering if it is correct/rigorous.

To complement densep's answer, here is a schematic Edgeworth box illustration of what can go wrong. The points on the dashed line are convex combinations of the Pareto optimal points $(x_1,x_2)$ and $(\hat{x}_1,\hat{x}_2)$, but the marked point is not Pareto optimal.


What are some important mathematics results that were first developed in Economics? Economics is known to import mathematical methods that were proven to be useful in other areas.

There is any important result in mathematics that was first developed in the context of Economics research and then explored in other applied fields such as Physics or Engineering?

The Envelope theorem is arguably one, https://en.wikipedia.org/wiki/Envelope_theorem . I would say Blackwell's papers on comparisons of experiments are a significant mathematical contribution--extending Hardy,Littlewood and Polya's earlier results, and paving the way for further investigation.

Econometricians have made statistical contributions as well; see e.g. https://en.wikipedia.org/wiki/Heckman_correction .

More recently, I saw a paper presented last year on how a certain type of stochastic PDE could be solved by reinterpreting it as a (stochastic) differential game. I would expect to see more related to this (the relationship between differential games and stochastic pdes) in the coming years, especially given the recent surge of interest in mean-field games.


Why don't Governments do away with the optics of taking on debt against new currency, and instead issue a limited currency every year (say 5% of GDP)? Governments engage in the optical illusion of taking on debt against issued currency.

The debt is effectively owed by the Government to itself.
The value of the debt is completely controlled and defined by the Government in realtime (insofar as it is denominated in the currency defined by the Government in realtime).

Each of the above two points independently are enough to render the debt meaningless.
What would happen if a Government decides to do away with the optical illusion of issuing debt against currency, and instead issue a limited amount of currency every year (say 5% of GDP) to finance its spending?
[I understand that there would be inflation, but that happens regardless of whether or not the Government takes on this cosmetic debt against issued currency.]

Well first of all we have to make it clear that we are talking about government debt held by central banks. Non-trivial amount of government debt will be borrowing from households firms and other governments where it is not valid to argue that the debt is an illusion.
When it comes to debt held by central banks not much would change although it would make reducing amount of money in circulation bit difficult for a central bank. An advantage of creating new money by purchasing government bonds is that the money can be easily destroyed just by central bank asking for government to pay back the money. In principle government has other options of how to reduce money supply even in such case but doing it through repaying bonds is the most direct way how central bank can do it without help of rest of the government.
This brings me to the final reason why such institutional arrangement is justifiable in the first place. It is virtually agreed by most economists that central banks work best when they are independent technocratic institutions rather than when they are directly controlled by executive branch of the government (and consequently under political influence). You can read about all the benefits of central bank independence in this ECB working paper, but as a non-economists you can view it analogously to courts independence. You want law to be administered by experts in impartial manner that will help society run more efficiently than political courts and similarly independent monetary policy helps more efficient management of money supply and other things that are under purview of monetary policy.
Having central bank and rest of the government operate at arms length, like increasing money supply through purchase of bonds, or by not even buying bonds directly at primary market but only at secondary market helps to build a wall between government and central bank.


How to understand the USDA Demand/Supply report ? Need you help ..
Have couple questions about the Wheat Report

M Acres stands for ? Million acres ?
Yield Bu/Acre , can anybody explain this calculation ?
Stocks use ratio % , how the calculation works ?

M Acres is million acres, as you suggest
Bu is bushel, in theory a volume of 8 US dry gallons, but apparently for wheat at 13.5% moisture it represents a weight 60 pounds.  Bu/Acre is therefore an average yield: with some minor rounding $\frac{2309}{43.9}\approx 52.7$ in the first column
Stocks/Use ratio seems to be the stocks at the end of the period divided by the total use (domestic and exports) in the period so in the first column $\frac{1181}{2222}\approx 53.2\%$


Do mainstream economic models use the blank-slate view of humans? Human capital is an important part of GDP forecasting. Do the models in use today view humans as "blank slate", meaning they can be educated to the same level, or do they take into account heritability of human abilities?

Standard economic treatments of skill generally assume that people have some baseline level of skill (human capital) which varies over people, and also that skill (human capital) can be improved via education or other methods.

Most of the time, models don't specify why baseline skill varies over individuals, but heritability would presumably be one reason. There are certainly some generational or evolutionary models that assume that parent and child baseline skill levels are correlated, which is in effect a heritability assumption.

As for GDP forecasting specifically, most GDP forecasting methods focus more on education level as the primary (or only) measure of human capital (both the average level and its distribution), but this is likely because baseline skill can't be measured very well, and can be proxied pretty effectively by education anyway because of selection effects.

So, in short, no, economic models aren't blank-slate, although using education as the only measure of human capital, which is sometimes done for pragmatic reasons, does look a tiny bit like a blank-slate assumption.


Difference equation in OLG framework This question is also regarding Aiyagari and Gertler (1985). We have the governments flow budget contraint:

$\frac{B_{t-1}}{p_{t}} + \bar{g}w = \tau_{y}(t)+\tau_{o}(t) + \frac{M_{t}-M_{t-1}}{p_{t}} + \frac{B_{t}}{(1+i_{t})p_{t}}$

Each period the government consumes the fraction $\bar{g}$ of the total endowment. $\bar{g}$ is i.i.d with mean $\bar{g}$. The government's expenditures are financed  by lump-sum taxes, money and one-period discount bonds. $M_{t}$ and $B_{t}$ are the supplies of money and nominal bonds at the end of period $t$.

$\tau_{i}(t)$ is the lump-sum tax where $i$ is either the old or young generation.

The authors then describe a fiscal policy rule:

  Assume taxes are levied only to meet the obligation
  on the debt. Further, let $(1 - \delta)$ be the fraction of the obligation which is backed by direct taxation, where $0 \leq \delta \leq 1$. That is, the present value of the
  stream of direct tax levies equals $(1 - \delta)$ times $\frac{B_{t-1}}{p_{t}}$

In and of itself I don't have any problem understanding this. However, they then define a time-stationary tax policy which satisfies:

$\tau_{y}(t)+\tau_{o}(t) = (1-\delta)[\frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}}-\frac{B_{t}-B_{t-1}}{p_{t}}]$ (2.8)

They describe this as:

  The policy requires that, each period, tax levies equal $(1 - \delta)$ times the
  difference between the present value of the current interest obligation on
  the debt and a term which corrects for the adjustment in the value of the
  obligation.

This I don't quite understand but I re-arranged the term inside the brackets and obtained $[\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}}]$

This makes somewhat more sense to me because I interpret it as: Each period taxes have to back $(1-\delta)$ the value of the outstanding real debt minus the present value of the future debt (I don't quite understand why the present value of next periods debt is subtracted...)

The next part is in regards to difference equations (I think). They define the expected present value of discounted taxes as $T_{t}$ where $T_{t}$ must satisfy:

$T_{t} = \tau_{y}(t)+\tau_{o}(t) + \frac{E_{t}(T_{t+1})}{(1+i_{t})E_{t}(p_{t}/p_{t+1})}$

I understand what the equation is saying but since I have barely done any dynamic programming at university yet I can't seem to solve it. The authors say:

  In view of (2.8) it is clear that $T_{t} = (1-\delta) \frac{B_{t-1}}{p_{t}}$.

I understand why this must be true (given the definition of the fiscal policy) but I cannot show this from the difference equation above.

I know one method of solving difference equations is to 'iterate forward' but I don't think I am doing it correctly.

So my question is regarding how to solve equations like the one above. And what should I read in order to get a better grasp at difference equations. I have heard Sargent and Ljungqvist is a good place to start. And some clarification regarding what the time-stationary policy means intuitively, would be greatly appreciated.

The following approach seems to work in this case:

look for the stead state formula of $T_t$. You can do this by taking (2.8) and the formula for $T_t$, and combine them. Then, get rid of all $t$ index:

$$ T = \frac{1-\delta}{p}\left(B - \frac{B}{1+i}\right) + \frac{T}{1+i}$$

Then, rearrange until you get $T$, the steady state formula for $T_j$. You should get:

$$ T= (1-\delta)\frac{B}{p} $$

Guess what the time indexes are, and confirm they are correct. To do this,  lag the equation for $T_t$ (so you get rid of expectations), and replace $T_{t-1}$ and $T_t$ with your guess. If the equality holds, your guess is correct. If not, try again. For example, let us assume  $T_t= (1-\delta)\frac{B_{t-1}}{p_t} $ (the correct one).

Then, the equation is:

$$ (1-\delta) \frac{B_{t-2}}{p_{t-1}} = \frac{1-\delta}{p_{t-1}}\left[B_{t-2} - \frac{B_{t-1}}{1+i_{t-1}}\right] + \frac{T_t p_t}{(1+i_{t-1})p_{t-1}} $$

After simplifying and rearranging you get:

$$T_t= (1-\delta)\frac{B_{t-1}}{p_t} $$

which is correct.

Now, I don't know how general this method is. This is not my area of knowledge. But at least it works here.


Stackelberg with 3 firms I'm currently trying to solve the following problem:

  Stackelberg with 3 firms Imagine there are three firms on a  monopolistically competitive
  market. The marginal cost of produc- tion in each firm is c. The
  demand is $p(q) = A − Bq$. The leader makes a production decision $q_1$,
  then two followers make a simul- taneous decision about their
  production levels $q_2$ and $q_3$. Calculate the quantity produced by firms
  in this economy, and compare it to Cournot outcome with 3 firms and to
  Stackelberg outcome with 2 firms.

My Workings

I've tried to solve the problem using the following method:

Leader ($q_1$)

$$\begin{align} profit(q_1,q_2,q_3) &= (A - B(q_1+q_2+q_3))q_1 -cq_1 -F \\
& = Aq_1 - B(q_1+q_2+q_3)q_1 -cq_1 -F\\
& = Aq_1 - Bq_1^2 - Bq_2q_1 - Bq_3q_1 - cq_1 -F\end{align}$$

The I took the derivative in regards to $q_1$ leaving me with this:
$$profit(q_1,q_2,q_3)= A - 2Bq_1 - Bq_2 -Bq_3 -c$$

Finally I just tried finding $q_1$:

$$q_1 = \frac{A-C-Bq_2-Bq_3}{2B}$$
To put $q_1$ into the equation and solve it for $q_2$ and then $q_3$

The Problem:

The begining of the solution in the answer sheet looks like this:

  Solution For Stackelberg with two followers, after firm 1 made its
  move, agents 2 and 3 are making their move simultaneously knowing q1.
  So, both firms 2 and 3 maximize

$$profit(q_i) = (A−B(q_1 +q_2 +q_3)−C)q_i ⇒ q_2 = q_3 = \frac{A − C}{3B} − \frac{q_1}{3}$$

Question:
as you can see my workings look nowhere near the answer, I've tried solving the problem using there method but I don't really understand were the $3$ in $3B$ comes from? and why are we using $q_i$? (also I don't quite understand why the method I used is incorrect)

Thank you very much for your help!

Start with the second stage, this is just Cournot competition between firm 2 and firm 3. You can solve this for the Nash equilibrium by setting the first order condition for firm 2 and firm 3 and solving these two equations, taking $q_1$ as given. This will give you quantities $q_2$ and $q_3$ in terms of $q_1$ which you can then plug into the profit function of firm 1 and you can maximize (i.e. find which $q_1$ firm 1 should choose to make sure the Nash equilibrium in stage 2 will be the most favourable Nash equilibrium possible for firm 1). So, step by step:

Start with the second round, find the Nash equilibrium by solving the following two equations:
\
$$\pi_2'(q_1,q_2,q_3)=0\\
\pi_3'(q_1, q_2, q_3)=0$$
Which will give you:
$$q_2=q_3=f(q_1)$$
plug this into the profit function of firm 1 and maximize this expression which has $q_1$ as choice variable.


Quantitative Marxian/Marxist micro and macro economic models? Are there quantitative Marxian/Marxist/neo-Marxist economic models like Dynamic Stochastic General Equilibrium models or Euro Area Wide (micro-founded) models - that can be used by central banks and researchers? Are there efforts to model relations between notions in Marxist economy? Maybe there is no need for separate economic models between Marxist (heterodox) and mainstream economics - maybe Marxists assume that mainstream economic models describe the reality quite accurately and that Marxism can add only decisions about resource redistribution and there is no need of models for such decisions.

I have found only one book and I have found some references about empirical work by Marxian economists - just compilation of statistics.

I wonder why I can not find more quantitative Marxism models and literature, because there should be such work:

Economists of communist countries (if we can say that - like China) and central banks of communist countries should have some models for making investment decisions, extensive models for planning and decision making (e.g. Greenspan remembered very impressive matrix calculations). Maybe these models are expression of Marxist economy.
Many countries after WWII experienced elements of state control, state planning and state intervention in economy (even US before till the Ford administration). So - these countries had to have some special models for planning and maybe they incorporated Marxian content?
I can not understand how it is possible to write about economics without using math and without having models. Marxian economists, seems to me, have this ability.

Any references, mentoring, pointers to the literature or important terms and notions are welcome!

Maybe one approach to finding my answer is using some workaround way - e.g. maybe I should seek the Chinese economic literature - it can have formalization of Marxism and at the same time current Chinese economic literature may be quite rational and quite acceptable for mainstream economics. Are there any resource of Chinese economic research in English?

There is a very interesting paper by John Roemer, published in Econometrica in 1980, that presents a mathematical general equilibrium Marxian model. I am not sure someone have ever estimated it though. Find it here (subscription required).
General equilibrium models (oftentimes idyosincratic) were also developed in the USSR. Check here for an account of them.
Models also informed (or attempted to inform) policy in USSR. The 1985 book "Macromodels of the National Economy of the USSR" gives an account of such models and policy applications.


Question about violating the zero conditional mean assumption Question for y'all with a background in econometrics.

Does the zero conditional mean assumption rely on complete randomness in a trial when doing regression analysis?

For example, if I was testing the effect of a free training program on wage, and the men in the trial were randomly given x amount of months of training, but the women were not assigned randomly, would that be a violation of a ZCM? I am struggling to make the connection between randomness and ZCM at all, but it was pointed out to me by someone that it is in fact influential.

If you introduce any sort of correlation between the explanatory variables $X$ and the error $\epsilon$ then the zero conditional mean may be violated. In your example, if you were to assign women their training time according to their age, then the exogeneity assumption breaks.

More formally, this last condition means

$$
\mathbb{E}[\epsilon|X] = 0
$$

If, wlog we assume $\mathbb{E}[\epsilon] = 0$, and there's a non-vanishing correlation between $X$ and $\epsilon$ then

$$
 0\not= \mathbb{C}{\rm ov}[X,\epsilon] = \mathbb{E}[X^t\epsilon] + \mathbb{E}[X] \mathbb{E}[\epsilon] = \mathbb{E}[X^t\epsilon]
$$

therefore $\mathbb{E}[\epsilon|X]\not = 0$


How does the GDP multiplier work? When we learn about the GDP multiplier, it is portrayed as 1/MPS, because savings are supposedly a leak out of consumer spending. However, what I don't understand is aren't those savings spent anyways in the product market? Don't they go to the financial market and get used by businesses to invest in projects? At that point the GDP multiplier would be infinite and the only thing that would matter is how fast people spend. Am I wrong in thinking this?

The multiplier comes from the solution to the goods market equilibrium. In economics everything is endogenous. Increase in income increases consumption that increases demand, demand increases production and production increases income. However, as an echo in a cave the initial increase in income gets 'weaker' as it cycles through the economy and the result 1/MPS tells you how much after accounting for the above mentioned cycle 1e increase in for example government spending increases the output.

Actually on fundamental level its not about marginal propensity to save but marginal propensity to consume (MPC), (however note that MPS=1-MPC) which is the channel through which increase in income affects the consumption which then affects demand and output.

A standard undergraduate textbook level consumption function looks something like this:

$$C = c_0 + c_1 (Y-T)$$

Where $C$ is the consumption, $c_0$ autonomous spending $c_1$ is the MPC and $Y-T$ is the disposable income ($Y$ is the income/output and $T$ taxes).  Mathematically, $c_1$ or MPC is what determines how $C$ changes when income changes. However, in order to see how much 1e increases output we cannot look at just MPC because as mentioned in the first paragraph the 1e cycles in the economy. So you actually need to calculate the geometric sum of the consumption multiplier being applied over and over again. Mathematically:

$$1+c_1+c_1^2 + ... c_1^n = \frac{1}{1-c_1}$$

Even though the 1e can be thought of circulating in economy infinitely  many times that does not mean it increases the output by infinity as many infinite sums actually converge to some actual value. In this case as long as $c_1$ is less than one, which it should as its the proportion of consumed income, the infinite sum would converge to some finite value not diverge to infinity.

However, while the above explains the logic behind why the multiplier is what it is it probably does not answer your question about saving. To see what role savings plays in the model consider the standard goods market equilibrium which is found by first substituting $C$ into the GDP identity and then solving for $Y$ (which is where you actually get the multiplier from):

$$Y = \frac{1}{1-c_1} \left(c_0  +I+G-c_1T\right)$$

Where as mentioned above $Y$ is output $1/(1-c_1)$ is 1/MPS $c_0$ autonomous spending, $I$ investment $G$ government spending $c_1 T$ are taxes multiplied by MPC. Now saving actually hides in the model in the investment $I$. By definition investment must be the sum of private and public saving, assuming away public saving for the sake of simplicity we can say that investment is equal to private saving so $I=S$. Consequently the multiplier 1/MPS affects also saving and when you increase the amount of savings the same multiplier is applied as to government spending or autonomous consumption so you are completely correct in wondering about that.

However, the reason why 1/MPS 'the multiplier' is that, as already mentioned, in this model the channel through which multiplier works is the effect of income on consumption which then affects all the rest.

Also note that in a short run $I$ can be considered fixed $\bar{I}$, I wont go into full detail about why that is as it is outside the scope of your question but basically in short run during recession investment is not always responsive to increase in savings because any increase in savings will be offset by changes to other variables. A consequence of assuming that investment is fixed and does not change with the level of savings then the savings will have no multiplier and you will get the "paradox of thrift".


Why is the equilibrium price not anything between \$0 and \$200? Consider the second problem below from Varian’s Intermediate Microeconomics:

  Suppose that there were 25 people who had a reservation price of \$500, and the 26th person had a reservation price of \$200.

  In the above example, what would the equilibrium price be if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?

For 24 apartments, the equilibrium price is \$500, and for 25, it’s \$200–\$500. So far, the answers in the back agree. However, for 26 apartments, I think the the equilibrium price should be \$0–\$200, whereas the book just says \$200 flat. I draw the supply-demand curve like this, where the blue line represents supply and black is demand:

Here’s a screenshot of the answers for reference:

Is the textbook correct, and if so, what am I missing here?

Varian is indeed wrong here in that there is a critical omission: He should have specified whether there in a 27th person and what this person's reservation price is.

If there is no 27th person, then the set of possible equilibrium prices is $[\$0,\$200]$, as you correctly point out. Explanation:

At each $p\in(\$0,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$200$, we have $S(p)=26$ and $D(p)=\{25,26\}$, so that there is a possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=26$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

If there is a 27th person and her reservation price is $k<\$200$, then the set of possible equilibrium prices is $[k,\$200]$. Explanation:

At each $p\in(\$k,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$k$, we have $S(p)=26$ and $D(p)=\{26,27\}$, so that there is a possible equilibrium.

At $0<p<k$, we have $S(p)=26$ and $D(p)=27$, so that there is no possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=27$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

Notes.

I assume throughout, as I believe Varian has, that each apartment-owner's reservation price or willingness-to-accept is \$0. I also assume that negative prices are impossible.
Following usual practice in economics, I've been a little sloppy about whether the elements of the codomains of the supply and demand functions $S$ and $D$ are numbers or sets of numbers. (Technically and most generally, they should be sets. But so as not to confuse students, we usually and more simply say they are numbers.)


What is the difference between the demand schedule and the demand curve? What is the difference between the demand schedule and the demand curve that a monopolist or monopsonist has? Could someone provide details for both cases or even some paradigm or a graph. I can not clarify this concepts.

Demand schedule is a table that gives you the quantity demanded at different prices.

An example of demand schedule that I found on Wikipedia is shown below:

Demand curve is a curve that plots the demand at different prices in the 2D space defined by $Q$ and $P$ (see example picture I took from investopedia below).

Demand curve is essentially a plot of demand schedule.


Rigorous proof needed: Acemoglu (Intro Growth) Corollary $2.1.2$ I am reading Acemoglu's intro to modern economic growth. But I am having trouble understanding his proof to a theorem related with stability. Here is the theorem:

And here are some related definitions and theorems:

My question lies in his proof of corollary $2.1$. How did he show that the sequence $x(t)$ is monotone and bounded above by $x^*$? I think it is not enough to use that the derivative is less than $1$ and the distance to $x^*$ is shrinking to say $x(t)$ is monotone. Can someone give me some more details to understand the proof?

Thanks!

The OP correctly identified a mistake here. Since the author claims monotonicity for a general function, let's disprove it for the simple linear case.
Consider

$$x_{t+1} = g(x_t) = -0.5x_t$$

This satisfies all the requirements and conditions stated by the author, and also that the derivative is smaller than unity in absolute values $\forall x$. But it converges with "damped oscillations", and so it is not monotonic since, say,

$$x_0 = -3 < 0 = x^*$$
$$x_1 = 1.5$$
$$x_2 = -0.75$$
etc


HP Filter Smoothing Parameter Mueller (2015, working paper) says that an HP filter with smoothing parameter 900,000 (for monthly data) corresponds to a smoothing parameter of 100,000 for quarterly data.

How does one do this exact calculation? If I were to calculate the corresponding annual parameter, how would I proceed?

Morten O. Ravn and Harald Uhlig (2002)

  This paper complements these insights using two different analytical
  approaches. The first approach uses the time domain  and  focuses  on
  the  ratio  of  the  variance  of  the cyclical component to the
  variance of the second difference of the trend component: this ratio
  is often used for calculating the smoothing parameter. For a
  particular benchmark stochastic process, it is shown that time
  aggregation changes this ratio by the fourth power of the observation
  frequency. The second approach uses the frequency domain and
  investigates the transfer function of the HP filter, thereby obtaining
  a general result. Again, a change-of-variable argument shows that one
  should adjust the HP parameter with approximately  the  fourth  power
  of  the  frequency  change.  Both approaches therefore yield a value
  of approximately $1600 / 4^4 = 6.25$ for annual data, which is close
  to the value of 10 given by Baxter and King (1999).

Source: Notes On Adjusting the HP-Filter for the Frequency of Observations

Mueller's result is not obvious to me given this rule. By that rule of thumb, a monthly parameter  given an quarterly parameter of 900,000 is $900,000 / 3^4 \approx 11,111$ (because there 3 months in a quarter). It also gives a yearly parameter of $900,000 / 12^4 \approx 43$ (12 months in a year).

The following quote seems to support Mueller.

  On suboptimality of the Hodrick–Prescott filter at time series
  endpoints Hodrick and Prescott (1997) proposed, on somewhat subjective
  grounds, a value λ = 1600 for quarterly data. However, it is desirable
  to adjust this value when observations of different frequencies are
  subject to the filter. Backus and Kehoe (1992) suggested an adjustment
  of the value by multiplying the standard value of 1600 with the square
  of the frequency of observations relative to quarterly data. For
  example, the relative frequency is 3 for monthly data and 1/4 for
  annual data. Hence, the corresponding values of the smoothing
  parameter is $\lambda$ = 100 and 14,400 for annual data and monthly data,
  respectively. This suggestion has been also used in commercial
  packages such as EVIEW. We shall use these values throughout the
  paper. With regard to the choice of the smoothing parameter, it is
  worth noting that, in research that has gone largely unnoticed in this
  field, Akaike (1980), while further allowing a seasonal component in
  the decomposition, proposed precisely the HP approach together with a
  data-dependent Bayesian procedure for the choice of $\lambda$.

So just as $1,600 (quarterly) \rightarrow 3^2 \cdot 1,600 =  14,400 (monthly)$ so to would $100,000 (quarterly) \rightarrow 3^2 \cdot 100,000 =  900,000 (monthly)$

Misea, Kimb, and Newboldc (2005)

But the cited Backus and Kehoe (1992) result seems to be superseded by the Ravn and Uhlig (2002) result, so this may not reflect the state of the art thinking on this matter.


Historical US Housing Data Availability Papers like Ng and Moench (2011) and Negro and Otrok (2007) use historical housing data to estimate latent variables that correspond to some "true" or "truer" measure of house prices or housing activity. I've put together a table of series one could use as inputs in such an exercise. However, as we can see in the table below, very little data is available before the mid-1970s. I would like to expand this list with U.S. historical data (free or to purchase) that help to characterize real and financial housing activity as far back as possible.  What other series am I missing?

+------------------------------------------------------+--------------+--------------+------------+
|                        Series                        |    Source    |  Frequency   | Start Date |
+------------------------------------------------------+--------------+--------------+------------+
| Median Sales Price of Single-Family Existing Homes   |  NAR         |  Monthly     |  Jan1968   |
| Median Existing Home Prices                          |  NAR         |  Monthly     |  Jan1999   |
| Median Sales Price of Single-Family New Homes        |  CENSUS      |  Monthly     |  Jan1963   |
| Average New Home Prices                              |  CENSUS      |  Monthly     |  Jan1975   |
| S&P National Home Price Index                        | S&P          |  Quarterly   |  Q1-1987   |
| Freddie Mac House Price Index                        |  FHLMC       |  Quarterly   |  Q1-1975   |
| FHFA Purchase-Only Index                             |  FHFA        |  Monthly     |  Jan1991   |
| FHFA Home Prices                                     |  FHFA        |  Quarterly   |  Q1-1975   |
| CoreLogic National House Price Index                 |  CoreLogic   |  Monthly     |  Jan1976   |
| Housing Starts                                       |  CENSUS      |  Monthly     |  Jan1959   |
| Existing Home Sales                                  |  NAR         |  Monthly     |  Jan1999   |
| New One-Family Houses Sold                           |  CENSUS      |  Monthly     |  Jan1963   |
| Housing Units Authorized by Permit                   |  CENSUS      |  Monthly     |  Jan1960   |
| Housing Units Under Construction                     |  CENSUS      |  Monthly     |  Jan1970   |
| Housing Completions                                  |  CENSUS      |  Monthly     |  Jan1968   |
| Private Residential Fixed Investment (PRFI)          |  BEA         |  Quarterly   |  Q1-1947   |
| Wilshire REIT Total Return Index                     |  Wilshire    |  Monthly     | Jan1978    |
| Ziman-CRSP Residential REIT Value Weighted Index     |  CRSP-Ziman  |  Monthly     | Jan1980    |
| Ziman-CRSP Residential REIT Equally Weighted Index   |  CRSP-Ziman  |  Monthly     | Jan1980    |
+------------------------------------------------------+--------------+--------------+------------+

Commercial real estate data is not ideal, particularly data that is contaminated with mine and oil and water well construction. Regional data is fine but national data is preferred.

I feel this barely qualifies as an answer, but doesn't Shiller have house prices data on his webpage? Currently here.

Specifically:

  Historical housing market data used in my book, Irrational Exuberance [Princeton University Press 2000, Broadway Books 2001, 2nd edition, 2005], showing home prices since 1890 are available for download and updated monthly: US Home Prices 1890-Present.

I don't have much experience with this data directly, and am only aware of it through  Khandani, Merton, Lo (2013) ("Refinancing Ratchet Effect"), where they use this and many other data sources to construct a (very long) housing dataset for their simulation analysis.

Details of Case Shiller Series Construction methodology (from the end-notes of the 2006 edition for chapter 2):

  Even though there were no regularly published home price indexes
  before the 1960s, some economists were constructing indexes of home
  prices that cover most of the years since 1890. We found home price
  indexes from 1890 to 1934 and from 1953 to the present that used in
  their construction some device to attempt to hold the quality of the
  home constant.

  The nominal home price index 1890–1934 is from Leo Grebler, David M.
  Blank, and Louis Winnick, Capital Formation in Residential Real
  Estate: Trends and Prospects (Princeton, N.J.: National Bureau of
  Economic Research and Princeton University Press, 1956). It is a
  repeated-measures index based on a survey of homeowners in twenty-two
  U.S. cities, who were asked to give the value of their home in 1934
  and the date and price of the purchase of that home. Since it is based
  on repeated measures of individual homes, the series is protected, in
  contrast to the simple median price, from any bias from changes in the
  mix of houses sold or of the increasing size and quality of newer
  homes. Its shortcoming is that it depends on memories of the surveyed
  homeowners for the earlier purchase price.

  The nominal home price index that we constructed for 1934–53 is a
  simple average over five cities of median home prices advertised in
  newspapers. The cities are Chicago, Los Angeles, New Orleans, New
  York, and Washington, D.C. My students collected the data from
  microfilmed newspapers at the Yale University library, collecting
  approximately thirty prices for each city and year, except that for
  the fifth city, Washington, D.C., 1934–48, data came from a median
  price series from E. M. Fisher, Urban Real Estate Markets:
  Characteristics and Financing (New York: National Bureau of Economic
  Research, 1951). The median series for 1934–53 does not make any
  attempt to correct for home quality change, as do the indexes we use
  for the other subperiods. Improvement in home size and quality gives
  median home price an upward bias, and this is why I avoided using
  median price outside the 1934–53 interval.

  The nominal home price index for 1953–75 is the home purchase
  component of the U.S. Consumer Price Index (CPI). The Bureau of Labor
  Statistics collected data on home prices for those years for homes
  that are held constant in age and square footage. In the 1980s they
  discontinued this index when they switched to a rental equivalence
  basis for housing in the CPI. They made this change to correct what
  was considered a conceptual flaw in the housing component of the CPI:
  the CPI is supposed to be a price of consumption goods and services,
  not of investment assets. For our purposes, however, the old home
  purchase component is acceptable. There are, however, some
  shortcomings in the home purchase component, notably that it is based
  only on homes with certain government-subsidized mortgages, and the
  procedure that the Bureau of Labor Statistics used to correct for
  changes in the ceiling on these mortgages was not optimal. See J. S.
  Greenlees, “An Empirical Evaluation of the CPI Home Purchase Index
  1973–8,” American Real Estate and Urban Economics Association Journal,
  Carnegie-Rochester Conference Series on Public Policy, 17 (1982):
  203–38.

  The nominal home price index for 1975–87 is the U.S. home price index
  published by the U.S. Office of Housing Enterprise Oversight (OFHEO),
  which is available on their Web site. It is a repeat sales index, and
  thus controls for quality change. The nominal home price index for
  1987–2004 is the repeat-sales U.S. home price index produced by Fiserv
  CSW, Inc., successor to Case Shiller Weiss, Inc.

  Since 1987, the CSW and the OFHEO series have shown very similar
  patterns through time, though the sharpness of the increase is
  slightly more pronounced in the CSW series, an observation we
  attribute to the fact that our series is based only on actual sales,
  while the OFHEO series also uses both actual sales and appraised
  values as if they were sales. Appraised values are somewhat sluggish
  to respond to new market conditions.


Recommended tests do detect breaks in time series What tests do you recommend for finding breaks in the following time series: log of inward FDI, log of nominal GDP, log of outward FDI from 1980 to 2012?

For a multivariate regression:

If you're testing for a break at a known date, say at a historic event or change in policy regime, I would recommend that you use a Chow test (a special case of an F-test). This will give you an idea of whether the coefficients are constant on each side of the proposed break.

On the other hand, if you want to test for a break at an unknown date, you have three options. You could try a Quandt, Mann-Wald, or Andrews-Ploberger statistic. The choice is up to you, but I like Mann-Wald. To me, it's more intuitive than the others because it's basically the Mean Value Theorem evaluated using a Riemann sum of the F-test at each potential break. Of course, the asymptotic distributions necessary to use any of these methods are beyond my knowledge, but you don't need those if you're modeling in a nice software package with pre-loaded diagnostic tests.

Lastly, testing for "breaks" (as in multiple breaks) is a computational nightmare. You probably have deeper issues with your model (like omitted variables or some sort of misspecification) if you need multiple breaks.


What is the empirical price elasticity of demand for insulin? A lot of textbooks say that demand for insulin is perfectly inelastic. I don’t think this is quite true (and I am a type 1 diabetic). It’s clearly very inelastic, but not perfectly so. There are things you can do to reduce your insulin intake, e.g. eat less carbohydrates.
I’m curious if any studies have actually gone out and measured this…

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).


Impact evaluation of a policy when different individuals adopt the policy at different times Let there be a policy $T$ which can either be implemented or not implemented by an organisation. Suppose there are $n$ organisations and the outcome variable $Y$.
The policy is not centrally imposed- different organisations have adopted the policy on their own will at different points in time. Which suggests that there is a significant possibility of self-selection bias.
I wanted to know if there is any technique to find out if such a policy is effective or not.
I have read about Difference-in-Difference and Discontinuous Regression-- but as far as I have understood-- they need a policy implemented at a particular point in time, and two groups, one of which is treated and the other is not.
In my case, there are two groups-- one that is treated and one that is not treated, but the time of implementation is different for different organisations.
I may be unclear, so I present a simplistic example. Suppose there are 6 firms A, B, C, D, E and F, having all walls in their respective offices white or green. The $Y$ variable of interest to me is the total productivity of employees (suppose I have a way to measure that perfectly) for example.
In the beginning (suppose in the year 2000) all the firms had white walls. Then in 2003, firms C and E changed to green. Then in 2005 firm B also changed to green. The other firms have not made any changes till now (say, 2010).
I have the employee productivity data (monthly) of all the five firms from 2000 to 2010 and also the data of when they switched from white to green.
Is their any method to find out if green walls have any advantage over white walls with this data? I am aware of the possible self selection bias (maybe the firms decide to change to green when they see an unusual drop in productivity etc.) Is there any way to overcome this and evaluate if the policy of switching to green works?
If yes, where can I read more about it, the assumptions, how to test them, then how to actually conduct the analysis using software etc.

The description is bit vague  on how many organizations we are talking but you could still do differences-in-differences (DiD) for all treatment-control-pairs that adopted it at the same time  separately and then running meta-analysis on the results.
As far as I understand based on Angrist & Pischke (2009) Mostly Harmless Econometrics ch.5, DiD solves any possible self-selection bias as long as the identifying assumption of common trend is satisfied. That is in DiD it it is required for both treatment and control to follow common trend pre-treatment. If the changes in outcome variable ($y$) pre-treatment are virtually same between treatment and control, then even if the levels of $y$ are different the estimates won't be biased. This of course will have to be tested in each case.
The above will however only be feasible if you do not have to run too many of the individual DiD otherwise it would be unpractical.
An alternative could be the event study design (see MacKinlay, 1997) that is used quite often across finance literature. It allows for events to occur at different times and it does not matter that firms/organizations experiencing the event are heterogenous and possibly self-select for the event (this approach is often used to evaluate M&A effect on stock returns for example), but on the other side this approach is very data intensive and depending what precisely is $y$ it might not be appropriate, or also if there is event clustering.
In order to implement regression discontinuity design (RDD) there must be some cut-off point that determines whether treatment is assigned to subject or not. For example, RDD would be appropriate if there is a policy that requires any power plant producing more than let's say 1.21 gigawatts is required to adopt some $CO_2$ curbing technology and below that threshold it is not required, and you want to examine the effect of such policy on some outcome variable like air quality. There is also so called fuzzy RDD which does not require sharp cut-off in some deterministic variable as long as probability of adopting the treatment is significantly different  (see ibid. (2009) Mostly Harmless Econometrics ch.6). The feasibility of this will again depend on data availability and other details, but if the organization themselves have full control over probability of treatment it will not be appropriate.
Lastly, as you could have probably already guessed from the discussion above the answer the choice of methodology is always context dependent. In order for anyone to give you concrete answer you would have to share virtually every single detail of the study which is not feasible, so take the above as some pointers with references (since you tagged the question as a reference request) rather than telling you exactly what to do.


Bertrand Duopoly Equilibrium for Discrete Prices There are two identical firms, $1$ and $2$, with zero marginal costs. They produce homogenous product, which is demanded by a unit mass of identical consumers, each of which has inelastic unit demand with a reservation price of $2$. Prices are constrained to take only integer values. Using standard game theoretic reasoning, determine whether or not each possible pair $(p_1,p_2)$ can be regarded as Bertrand Nash equilibrium.

Would the Nash equilibria be $(p_{1}^{*},p_{2}^{*})=$ $\{(0,0),(1,1)\}$.
For symmetric prices, $\{(2,2)\}$ since the demand would be $0$, I can see how unilateral deviation to $1$ could increase profits for a firm.
However for prices $\{(0,0),(1,1)\}$, no unilateral profits are possible. For other non symmetric price pairings, unilateral deviations are possible. Therefore, the Nash equilirium prices are $\{(0,0),(1,1)\}$.
Is my reasoning correct?

The problem formulation admits the following Normal Form representation. We can reject any strategy involving price greater than 2, as demand falls to zero and such strategies are strictly dominated by those for which prices are either 1 or 2.

    0         1         2

0 [0,0]     [0,0]     [0,0]

1 [0,0]   [0.5,0.5]   [1,0]

2 [0,0]     [0,1]     [1,1]

With this setup you can see that there are three Nash equilibria: (0,0), (1,1) and (2,2). At (0,0), firms are indifferent to the outcome of a higher price, and will not deviate. At (1,1), Firm 1 will not deviate to 2, nor will Firm 2 deviate to 2, as this cedes the market to the opponent. At (2,2) firms are indifferent to reducing price to 1.

(0,1), (0,2), (1,0), and (2,0) are strictly dominated by deviating to a price-matching strategy, splitting the market between firms.

Similarly, (1,2) and (2,1) fail to be Nash equilibria because whichever firm opens with a price of 2 can capture half the market by deviating to a price of 1.


Book recommendations: Introduction to economics I recently made my first trip to a foreign country. I'm quite amazed by how things work. Why are things priced differently at different places? Is the bed-sheet that I bought from that country worth the same in my country? I brought my bed-sheet there in \$40 while the same will cost less than \$10 in my country.

Also, I learnt about utility through this video. I'm just in an awe. These things are so beautiful yet not visibly understandable to me. I'm sad as well as happy I didn't learn about any of these in my school. Sad because I potentially wouldn't have ever studied it. Happy because I can learn this thing with interest, as the school and university I attended have usually taught me in a way that makes things dull.

My background is bachelors in Computer Science. I'm decent at maths. I would like to ask you for book recommendations that will help me get some idea of how things work from an economic standpoint. How banks work, how loan works, what is money etc.

I would like you to suggest me two kinds of books:

That are example based, easy, concise, less mathematical but more story based.
That are mathematical and cover these theory in details.

Thanks.

It's great that you're developing an interest in economics. I would suggest Mankiw's Principles of Economics to start with. I believe it meets both your requirements and covers the two major areas of economics, microeconomics and macroeconomics, so you would get a decent overview of this field of study. Good luck!


Is "technical chart analysis" for stock development something that is scientifically supported? I assume the answer is "no" because if you can predict the development of a stock then people would get very rich. However there are so many YouTube channels and web sites where people talk about things like Elliot Waves and other theories.

There are not a lot of studies on this topic.  There were studies in the 70s that would tend to disconfirm the idea of technical analysis as it was proposed at the time.
The studies generated random sequences of numbers in a time series and had technical analysts then make decisions believing that the time series were real time series.  They didn't detect the substitution.  You cannot make systematic profits on a true random series.
Unfortunately, that is a weak falsification.  There is no evidence they got the time series correct. It may be the case that equity time series are different and that signals exist that are legitimate.
With that said, there is a bigger problem with actual time series.  Block orders are "taken off the tape," which means that block orders are not reported when they happen.  The time series is "as reported" rather than "as occurred."  Block purchases or sales are reported after the fact at the weighted average price.
It is not unreasonable to believe that the bid and ask do contain information even if the price record does not.  Whether it creates a "pattern," is another matter.
I am not convinced by the studies so far in either direction.  I do believe, however, that no retail investor should ever try it.  There is a giant information asymmetry between retail investors and institutions.  I don't believe that any retail investor can overcome the informational differences present in the system.
Indeed, during the tech bubble of the 90s, the State of Georgia did a study of 100% of all Georgia daytraders by looking at their statements.  They found that 90% of Georgia day traders lost 100% of their money and only one percent made a significant profit.

Arditti, F.D., McCollough, W.A. (1978), Can Analysts Distinguish Between Real and
Randomly Generated Stock Prices? Financial Analysts Journal, 70-74.

Barber, Brad M. and Terrance Odean, 2000, “Trading Is
Hazardous to Your Wealth: The Common Stock Investment
Performance of Individual Investors,” Journal of Finance
55 (No. 2, April), 773-806.

Bartlett, M.S. (1946), On the Theoretical Specification of Sampling properties of Autocorrelated Time Series, Journal of the Royal Statistical Society, Series B, 8, 27-41.

Grandia, V. (2002), The Search for the Golden Rule: A Reality Check of Technical
Analysis in the Dutch Stock Market, master thesis, University of Amsterdam.


Why are imported Chinese products in India cheaper than those locally produced? Why an imported Chinese product in India is cheaper compared to a locally manufactured item?

Consider an exact same product, say an umbrella, which is both imported from China in India and locally produced. The reason why the Chinese product is cheaper compared to the local one could be related to differences in productivity or total factor productivity. The Chinese product could be relatively cheaper because Chinese firms use better production techniques,
higher-quality general labor and capital input, better managerial practice/talent, or better information technology. See the JEL survey about What Determines Productivity?

The recent paper by Rodrik and co-authors documents that labor productivity is higher in China than in India (see Table 1). This could be a first reason why prices of the same product differ.


What are some helpful online tools (databases, etc..) for economic analysts? More specifically, I'm looking for a online database of US economic statistics that is regularly updated. Most of my searches result in older data, that is not particularly useful for what I need it for. I'm needing to make many cross references with it, for instance, comparing the unemployment rate of two cities/regions.

St. Louis Fed:
https://fred.stlouisfed.org/

Has very good US data.

I am not aware of a database that systematically aggregates city-level data. I recommend state government sites and, for larger cities, city government sites.


Bond pricing, why t=3? In question c-f. .Why do they calculate $t=3$? When they say $t=4$ at the beginning and then next year is $t=4+1=t=5$

If there are 4 years to maturity now, it will only be 3 years to maturity next year.
E.g., initial year is 2000, bond matures in 2004. In 2001, it’s 3 years to 2004.


Is Marginal Rate of Substitution a multivariable function? Suppose I have $U(x,y)$ and a level set of indifference curves. Suppose the value of $U$ along a given curve is $\bar{U}$. We know $dU = 0$. We compute total derivative, rearrange, and now have

$$\frac{dy}{dx} = -\frac{U_x}{U_y}$$

My Confusion and Question:

$\frac{dy}{dx}$ is written as a function of a single variable right? But when we talk about $MU$, we always talk about it as a multivariable function. I thought $MRS$ was the ordinal utility analog of the cardinal utility $MU$. So why is $MRS$ a single variable function?

$\frac{dy}{dx}$ is written as a function of a single variable right?

I don't think so. Your shorthand notation should be standing for

$$\frac{dy}{dx} = -\frac{U_x(x, y)}{U_y(x,y} $$

and hence is a function of both variables ${x,y}$. That makes sense, as your MRS could depend on the locus you're at.


If I fill a vending machine with \$0.50 sodas and sell them for \$1, am I creating 50 cents of wealth for the economy? This is a very simplified example.
If I fill a vending machine with sodas that cost \$0.50 each and someone comes along and pays \$1 for it, am I creating an extra \$0.50 of wealth for the economy? The wealth being that I've taken the beverage and delivered it to someone in a convenient format.
I am referring to the idea that the economy is not a zero sum game. If I buy something and sell it for a higher price we are not just shifting money around the economy but actually creating new money (assuming I provide some value in the process, but that is determined by the consumer).
So in the above example with the vending machine, does the federal reserve have to create an extra 50 cents of the currency in order to avoid deflation?
Another example: the company I work for generates an extra \$10m in sales this year than last year. So people in the economy have transferred \$10m from them to us. That equates to the creation of \$10m of wealth, right? And the fed just then issue another \$10m to keep the currency at the same level?
Note: I am a total laymen without any education in economics. These concepts - currency values, value / wealth creation - are simply of interest to me

You have to be very careful with your definitions here. Following the basic definitions (e.g. see use of terms in Mankiw Principles of Economics):
Value: Value depends on your marginal utility and typically it is the amount of money you are willing to give up for something.
Wealth: Is by the value of net assets.
Income: Is the net return to some activity.
You are right that economy is not necessary zero-sum game (although there are economic interactions which can be zero-sum), but your use of the term wealth is improper.
If someone pays you to fill the soda machine, you have costs \$0.5 but get paid \$1 your income will be \$0.5, you created at least \$0.5 of value because the person who paid you clearly valued the filled vending machine for at least \$1 otherwise they would not exchange that 1 dollar for your work, but at the same time you had 0.5 costs, so you are creating at least \$0.5 extra value.
The wealth is accumulated by saving (whether you save by buying house or putting money on your account value of your net assets increases). Hence if you have \$0.5 income and you save it all you also create \$0.5 wealth. If you consume half you only create \$0.25 wealth and so on.

So in the above example with the vending machine, does the federal reserve have to create an extra 50 cents of the currency in order to avoid deflation?

Here the answer is maybe. Inflation/deflation does not depend just on the value of output that you create.
Inflation/deflation is just positive/negative change in the price level which is in turn determined by the money market equilibrium. The money market equilibrium, in its simplest form is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing (so % changes in right hand side variables give us the % change in $P$) we get:
$$\ln P=\ln M + \ln V − \ln Y$$.
If you perform those extra services you increase real output by \$0.5, ceteris paribus, you are correct that Fed would have to create additional \$0.5 dollars to prevent inflation. But the, ceteris paribus, assumption might not hold in real life because velocity of money can change as well (plus in more complex models of money market equilibrium expectations play role and so on). Consequently, the correct answer here would be maybe. Under ceteris paribus assumption yes, but it is not guaranteed.

PS: Note sometimes even economists in casual speech equate wealth creation with value creation, so you might have heard some pundit or economists use similar example to argue wealth was created, but that is not how the two terms are rigorously defined in the literature.


Is Quantitative Easing an asset-swapping or money-printing exercise by the Federal Reserve? Shortly after 2008, there were prominent commentators such as Jim Rogers who called Ben Bernanke an idiot and condemn him for printing money. However, there were also articles that suggest quantitative easing is really an asset-swap and not money-printing. On the other hand, there are articles that say the opposite.

http://www.pragcap.com/bloomberg-quantitative-easing-isnt-printing-money/
http://www.bankingmyway.com/save/savings/what-does-printing-money-mean

What exactly is quantitative easing? Asset-swapping or money-printing? Does it raise the money supply? Inflationary or not?

Seignorage is harmful if it is abused to cause hyperinflation. Without quantitative easing, we probably would've had mass deflation instead (which is arguably worse), and makes the Fed look not credible in striving for its target inflation rate.

There are different ways of implementing quantitative easing in other areas I think, but more or less, it is buying massive amounts of financial assets from banks and other financial institutions in order to lower their yield, which raises the price and increases the money supply. It is supposed to be inflationary, but QE is usually in response to when the short term interest rate approaches zero, and it becomes hard to do that.


Does lowering consumption help reduce inflation rates? My country has a whopping 25% annual inflation rate. As far as I know, high inflation implies higher consumption rates, but I was hearing lately that consumption is actually lowering, while inflation rates keep climbing. I had the feeling that lowering consumption could help reduce inflation, but apparently it doesn't always work like that.

Does lowering consumption help reduce inflation rates?

Lowering consumption, and increasing savings can increase net exports. Since savings increase relative to consumption, there is more money to lend out to the world (in an open economy), and less investment domestically. The decrease in demand for your currency due to the reduction in investment and rise in savings can lead to a reduction in the real exchange rate for your currency relative to other currencies. NX (net exports) = S - I (Savings - Investment). S = Y-C-G (income - consumption - government spending). As you can see, the level of consumption directly impacts savings. A decrease in consumption increases savings, which in turn decreases the real exchange rate for your currency, which can make imports more expensive, creating nominal price inflation.

Another determinant of inflation however is also the money supply, and this is the biggest factor. The central bank in your country has the largest impact on money supply. This is because Y (output) is fixed based on the factors of production. % change in M (money supply) + % change in V (velocity of money) = % change in P (price level) + % change in Y (output). That is (% change M + % change V = % change P + % change Y). As you can see, the price level directly has to increase proportionately to the change in the money supply that the central bank can manipulate. So even if consumption decreases, nominal inflation rates will rise if the central bank is expanding the money supply. That is how it’s possible to see a decrease in consumption, and still potentially see inflation.


Social mobility and income inequality I once read an argument by the libertarian economist Thomas Sowell against the conventional wisdom that rising income is bad because most statistics which show rising income inequality do not account for individuals moving between income bands over time.

His argument went something like this. Consider the average professional at the beginning of their career, for example a junior doctor ( the argument also seems to apply to non-professionals but in a less extreme way). They won’t be in the top income bracket when they begin working but over the course of their career they almost certainly will at some point. For this reason income inequality is not as significant an indicator of the actual economic situation for many individuals as it does not track how the incomes of individuals change, and often increase, over time. Furthermore, income inequality statistics don’t count people who don’t work, which includes a large number of wealthy pensioners. Therefore rising income inequality can even be thought of as a positive thing as it affords young people the opportunity to earn more, and thus actual leads to less actual inequality of wealth.

I couldn’t find a link to his exact words unfortunately but I suspect that he was making the point from the point of view to defend free-market capitalism.

My question is simply is is argument correct, and if not why not?

I am a physics student, so I’m possibly just not familiar with the most technical economic theory that might explain this, but I can’t figure out why this fact (that people move between income brackets over time) is basically never addressed either in political discourse ( eg the Labour Party are always talking about income inequality but I’ve never heard the Conservatives give as eloquent a response as Sowell) or the popular literature that I’ve read ( eg newspapers on the right and left, and books by modern economists such as Ha joon Chang and David pilling).

The argument is correct

Social Mobility and Income Inequality are simply two distinct phenomenons. Mobility is arguably the more complex one, and there are many different ways to define it. As an example, absolute mobility refers to someone earning more income in absolute terms (\$ per year), while relative mobility refers to someone earning more in relative terms (going from the 60th to the 70th percentile of the income distribution). Note that the latter could occur for constant personal income, if income of all others decline. Another major distinction is made between inter- and intra- generational mobility. I would attribute part of the reason mobility is not featured often in political discource to the ambigiuous meaning of the concept. Related to this, there is no generally accepted single indicator (like the Gini coefficient for income inequality) by which mobility could be measured.

The second reason is that measuring mobility requires repeated observations of income distributions for the same individuals, ideally over a long period of time. These kind of data are still rather rare and not available for all countries in the required quality.

The scenario you have in mind ("the guy is not poor because he is likely going to earn a lot") is essentially a prospective task where one would estimate future earnings paths. While this can be done in principle, the precision of such studies are far below e.g. a detailed analysis on poverty ratios for various demographic groups.

Summing up, measuring income inequality ignores the movement of individuals between two points in time, which should be kept in mind when talking about income inequality. Povery is much more a problem if it's not a transitory, but a permanent phenomenon from an individual perspective.


The sequence of Money Supply - Interest Rate Until now I've learned that the Fed controls the interest rates and that affects the money supply. For instance,

The Fed increases Interest Rate ----> makes firms/households reluctant to borrow and Banks loan less ---> less money supply.

However, what is the economic intuition for the opposite sequence? So

"Fed increase money supply ----> interest rate decrease"
"Fed decrease money supply ----> interest rate increase"

I have never thought of this sequence - I have always thought of the interest rate as the causal factor and the money supply change as the result. I would appreciate it if somebody could explain how the opposite sequence is also possible. So to put it simply,

if Fed increases/decreases the money supply, why does interest rate decrease/increase? not in economic theory, but in real-life explanation please.

It’s just supply and demand. The interest rate is the price at which you can borrow money. For a fixed demand curve, if there’s less money available (the supply curve shifts left), the market will clear at a higher price (higher interest rate). Similarly, if there’s more money available for lending, the equilibrium interest rate will be lower.


From Discrete to Continuous Time I know one form of a derivative, which is

$$ f'(x) = \lim_{\Delta\to 0} \frac{f(x+\Delta) - f(x)}{\Delta}$$

Let $x_{t+\Delta} = x_t + \Delta \dot x_t$. Achdou and coauthors (end of appendix A1) claim that

$$ \lim_{\Delta\to 0} \frac{f(x_{t+\Delta}) - f(x_t)}{\Delta} = \lim_{\Delta\to 0} \frac{f(x_t + \Delta \dot x_t) - f(x_t)}{\Delta} = f'(x_t)\dot x_t$$

I understand the first equality, that's merely the substitution. I can't wrap my head around the second one. How exactly does one show it?

$$\require{cancel}$$

Notice that if $\dot x_t \neq 0$, then $\Delta \dot x_t \to 0 \implies \Delta \to 0$

$$ \lim_{\Delta\to 0} \frac{f(x_{t+\Delta}) - f(x_t)}{\Delta} = \cancelto{f'(x_t)}{\lim_{\Delta \dot x\to 0} \frac{f(x_t + \Delta \dot x_t) - f(x_t)}{\Delta \dot x_t} \dot x_t}= f'(x_t)\dot x_t$$

If $\dot x_t = 0$, then $f(x_t)$ is a constant function.


What would happen if a country defaulted on its public debt? Could someone explain in full detail to someone that does not understand much what would happen if a country refused to pay its pubic debt?

Someone told me if this happens banks would fall and citizens wouldn't have jobs and savings, but I cannot link these two events. How are these events related? How does one thing lead to another?

A “full explanation” would be extremely long; depending on the situation, a great many things would happen.

For developing countries, defaults are typically on debt in foreign currency, issued under foreign law. They will end up in legal fights with bondholders, and/or end up dealing with the IMF (or other body) to restructure their debt. Economic problems will revolve around being cut off from the financial markets, such as the currency markets.

The scenario you are describing seems to more about countries that issue debt in their own currency, under their own law (for example, the United States, Japan, Canada, etc.)

For most of these countries, the government is a large borrower. If they refuse to make any payments on debts (which is rare; defaults are often partial), this would represent a loss that is comparable to one year’s production or income (GDP). This would represent severe losses for many savers.

Furthermore, banks hold government bonds to manage their liquidity. Meanwhile, they have very little equity when compared to the size of their balance sheets. Large losses on their bond portfolios would likely result in their bankruptcy. Their bankruptcy would then cause losses on their lenders - bank bond holders, depositors. Furthermore, a bankrupt bank cannot extend new loans, and so any businesses that survived would have a difficult time to borrow to fund expansion. As a result, there would be massive job losses.

In practice, defaults are rarely total (except after wars/revolutions),so we do not see the full potential effects. However, even partial defaults - such as in Greece after the Financial Crisis - typically result is a depression. (I suggest that you look up the Greek debt crisis for an example. The Greek situation is different than developed countries outside the euro, since it does not control the currency it is borrowing in, and so was forced to deal with supranational bodies.)


Is money a conserved quantity? Excuse the naïve question, but I have a rather limited understanding of economics, and this question has been bothering me since my school economics teacher couldn't answer it satisfactorily many years ago.  It is also a question that is a prerequisite to many others I have.

Is money a conserved quantity, in the same sense as we have conserved quantities in physics (e.g. energy)?

In daily transactions that I have experience with, it obviously is. If I buy something from Joe, and give him \$10, I will have \$10 less and he will have \$10 more.  But the total amount we have together has not changed.

Does this apply to all transactions?  If not, when is it violated?  In what situations is money created, and when does it disappear (other than physically destroying paper based money)?

This then leads to a closely related question: Where can I find out how much is there of a given currency?  How much USD exists in total, if we combine the dollars owned by everyone in the world?

If this question is considered too basic, I would appreciate suggestion about where a mathematically minded person might read up on this topic.

Νominal money could be seen as a "conserved quantity" in a similar/analogous sense that we have in physics, under some qualifications and restrictions.

I stress that we are talking about the nominal value of money, which is an indisputable quantity and legal tender (the concept of "real" value is an estimated value and hence subject to disagreements etc.)

Then if we exclude
1) The central bank/government with the authority of the law
2) The commercial banking system as regards Transactions that have to do with taking or repaying loans
3) Actions that just destroy the physical carrier (say, burn the paper bill)

...then transactions do not alter the amount of nominal money, and in that sense, the nominal value/amount of money is conserved.

We exclude 1) because the central bank / government have the power by law to create new money (and destroy existing money, for example "cutting zeros" after a hyperinflation episode).

We exclude 2) because commercial banks create money under the "fractional reserve banking" system, while money is destroyed whenever a loan is repaid (the part that has to do with the principal, not the interest). The OP should ask a different question if the OP wants to know about that.

We exclude 3) because we can argue that these are not "normal economic transactions".

So the system in which nominal money can be considered a "conserved quantity" is a subset of a real world economic system.


What would be the results of improved goods manufacturing efficiency? Improved manufacturing efficiency could cause price reduction of goods, increase manufacturer's profit (still asks for the same price, but has less expenses), and what else? Is it possible to predict what would be more dominant? If not, is there historical statistical data about this?

Step 1) Improved manufacturing efficiency would lower the marginal cost of production for firms, thusly lowering the average total cost per unit.

Step 2) In a generic competitive market, lower ATC leads to profits in the short run, and increased number of producers in the long run.

Step 3) Increased number of producers moves the supply curve to the right (increasing supply), and the resulting equilibria has a lower price and a higher quantity. Profits for the individual firms decline as price falls, settling again at zero.


When only half of the independent variables are non stationary, does it make sense to run a cointegration test? I have the following regression equation (panel data):
$Y = f(X_1, X_2, X_3, X_4)$

After obtaining CIPS and CADF statistics, $X_1$ results to be stationary for both intercept and intercept + trend, whereas $X_2$ is stationary only for intercept + trend. $X_3$ and $X_4$ are contrariwise non stationary in neither case.

Because of these results, I was wondering whether running a co-integration test makes sense (since if I'm not wrong, all the variables should be $I(1)$).

Also, I was wondering whether to proceed with using a first difference for $X_2$, $X_3$ and $X_4$, or only for $X_3$ and $X_4$.

Thank you very much.

Kodi

For cointegration analysis all variables should be of the same order of integration. However this being said it depends on how much you trust your stationarity tests. Most unit root tests are riddled with low power problems especially in small samples (see some textbook on econometrics like Stock & Watson introductory econometrics for more details and sources).

Moreover, if the broader literature suggest that all the variables are let’s say I(1) I would even disregard the unit root tests results as they may be just false positives/negatives.

Also, if you want you can use the Pesaran bounds approach to cointegration which allows for both stationary and nonstationary variables being estimated within one model, although cointegration can only happen between stationary variables. This approach is usually not in textbooks yet so you might wanna check the original paper “Bounds testing approaches to the analysis of level relationships”


Interpretation of lagrange multiplier A student wishes to minimize the time required to gain a given expected average
grade, , in her end-of-semester examinations. Let $\displaystyle {t}_{i}$ be the time spent studying
subject i ∈ {1,2}.

Suppose that the expected grade functions are $\displaystyle {g}_{1} ({t}_{1})=40+8\sqrt{{t}_{1}}$ and  $\displaystyle {g}_{2} ({t}_{2}) = 2{t}_{2}$.

Thus the individual's optimization problems is to choose ${t}_{1}$ and ${t}_{2}$ to minimize total studying time $ = {t}_{1} + {t}_{2}$ subject to obtaining a mean grade of  where

$\displaystyle m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}=0$

I need to solve for the optimal choices of ${t}_{1} , {t}_{2}$ and $$ in the case where the student wishes to obtain an expected mean grade of 70.

Working through the problem I end with

${t}_{1}=4$

${t}_{2}=42$

$T=46$

$\displaystyle\lambda=-1$

How would I interpret the lagrange multiplier in this case?

As mentioned in the other answer, the Lagrange multiplier is the marginal effect on the value (optimized) function, when the constrained is "relaxed" marginally. In your case then it should be interpreted "how much studying time changes as the required average expected grade..." increases?

Well, this is a good example to show that how we set up the Lagrangian function matters in order to interpret what we get meaningfully. Your is a minimization case so we have
$$\min T = t_1+ t_2 \\
s.t\;\;\; \displaystyle m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}=0$$

How do we write the Lagrangian? Do we write

$$\Lambda = t_1+ t_2 + \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

or

$$\Lambda = t_1+ t_2 - \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

Since the constraint is an equality constraint, you may hear that "it doesn't matter". Mathematically, it indeed doesn't, but it does matter when the time comes to interpret the value of the multiplier. And, maybe in Utility maximization problems we don't much care because utility is ordinal. But in a problem like the one the OP is solving, the objective function is measured in a very real and measurable and quantifiable unit, time.

Now how we should understand "what happens when the constraint is relaxed"? In the specific minimization problem, it intuitively means "what happens when the required $m$ is reduced". (Since "relax" has a "desirable" connotation, and so we would desire to be asked for less effort, a lower target).

I guess it is now clear how the obtained value for $\lambda$ can be interpreted. And do ponder what variant of the Lagrangian is the appropriate one for your case.


Panel data, simple rearrangement? Consider the Correlated Random Effects model $y_{it} = \alpha + x_{it}\beta + \bar x \gamma + w_i + \epsilon_{it} $ where $x_{it}$ is a scalar explanatory variable.

The correlated random effects GLS estimator $ \hat \beta_{CRE} $ is the OLS estimator of $\beta$ in the quasi-demeaned regression

$\tilde y_{it} = \delta + \tilde x_{it} \beta + \bar x_i \rho + u_{it} $,

where $\tilde y_{it} = y_{it} - \theta \bar y_i , \tilde x_{it} = x_{it} - \theta \bar x_i $ and $\theta = 1 - (\sigma^2_\epsilon/ (\sigma^2_\epsilon + T\sigma^2_w))^{1/2} $

Question: I need to show that the residuals from the regression of $x_{it} - \bar x_i $ on a constant and $\bar x_i $ is just $x_{it} - \bar x_i $ itself.

Attempt: Regress $x_{it} - \bar x_i = \alpha + \bar x_{i} + \tilde r_{it} $ rearrange to get the residuals, $\tilde r_{it} = (x_{it} - \bar x_i) - (\alpha + \bar x_{i})$ I'm not sure how to proceed.

By stnadard OLS regression results, in the simple regression

$$y_t = \alpha + \beta z_t + v_t, \;\;\;t=1,...,T$$

we have that

$$\hat \beta = \frac {\sum_{i=1}^T (z_t - \bar z)(y_t-\bar y)}{\sum_{i=1}^T (z_t - \bar z)^2}$$

and

$$\hat \alpha = \bar y - \hat \beta \bar z$$

So the residuals are

$$\hat v_t = y_t -(\hat \alpha +\hat \beta z_t)$$

Then, substitute for your particular $y_t$ and $z_t$ in all the above.