Making mathematical sense of the expression for realized bond return I came across the following statement regarding the realized 10-year maturity bond's return over a year:

  The realized bond return (H) over a year has two components: the yield income earned over time and the capital gain or loss due to yield changes:
  $$H_{10} \approx Y_{10}-\text{Duration}_{10} \times \Delta Y_{10}.$$

I am a complete economics rookie and I'm trying to understand what's going on here from the mathematical point of view, which I'm gonna present here, but my calculations don't seem to add up.

If we denote the bond's coupon with $C$, and the bond's time $t=0$ yield to maturity with $y_0$, then the bond's value at time $t=0$ equals:
$$V_0=\frac{C}{1+y_0}+\frac{C}{(1+y_0)^2}+\dots+\frac{C}{(1+y_0)^9}+\frac{F+C}{(1+y_0)^{10}}.$$

At time $t=1$, we can express the bond's value as the function of the time $t=1$ yield to maturity $y$, so we have
$$V_1(y)=\frac{C}{1+y}+\frac{C}{(1+y)^2}+\dots+\frac{C}{(1+y)^8}+\frac{F+C}{(1+y)^{9}}.$$
Derivative of $V_1$ with respect to $y$ is equal to:
$$\frac{dV_1}{dy}=-1\cdot\frac{C}{(1+y)^2}-2\cdot\frac{C}{(1+y)^3}-\dots-8 \cdot \frac{C}{(1+y)^9}-9 \cdot\frac{F+C}{(1+y)^{10}} .$$
Now, we can apply some basic calculus here and state that for $\Delta y$ small "enough", we have that
$$ V_1(y_0+\Delta y)\approx V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y. $$
So now, if we consider the absolute return on our position (buying this bond at time $t=0$, selling it at $t=1$) from the time $t=0$ perspective, under the assumption that the time $t=1$ bond's yield to maturity is $y_1=y_0+\Delta y$, we have that:
$$\text{AbsReturn} \approx-V_0+\frac{C}{1+y_0}+\frac{V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y}{1+y_0}.$$
That is - we buy the bond for $V_0$, at the end of the first year we are paid the coupon which discounted value is $\frac{C}{1+y_0}$, and the approximation of the time $t=1$ bond's value taking into the account the YTM change is $V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y$ and we also discount it to time $t=0$.

Now, we can simplify the expression for AbsReturn since $-V_0+\frac{C}{1+y_0}+\frac{V_1(y_0)}{1+y_0}=0$ and we get:
$$ \text{AbsReturn}= \frac{\frac{d V_1}{dy}(y_0)\cdot \Delta y}{1+y_0} ,$$
which I guess we can also divide with our initial investment of $V_0$ to get the rate of return so we get:
$$ \text{RateOfReturn}= \frac{\frac{d V_1}{dy}(y_0)\cdot \Delta y}{V_0(1+y_0)} ,$$
aaand this is where I completely lose it. I can't seem to understand the connection between the original expression and the thing I end up with. What does the term $\text{Duration}_{10}$ in the original formula even stand for - I guess it is the derivation of bond's value with respect to yield - but bond's value at what time: $t=0$ or $t=1$? Does it even make any difference? If it is at time $t=0$, how can we be using linear approximation of that function for approximating bond's value change at time $t=1$? I'm completely puzzled over this. Am I doing something completely wrong in this derivation? I appreciate any insights on this. Thanks!

The approximation given is off at a key point - it is the duration of the bond at 9 years maturity, not the original duration.

The approximation works in two steps.

If the yield is unchanged over one year, the total return over the year is roughly equal to the starting yield.
If the yield changes, we add a capital gain/loss which is equal to the change in yields times the yield sensitivity. This is the modified duration of the bond, at 9 years maturity.

This approximation ignores the reinvestment of any coupon income, and the convexity effect (the return sensitivity changes slightly as yields change). Also, the bond yield convention is probably different than the convention used for total return. That said, for yield changes under 50 basis points, the approximation is typically pretty good.


How does the IS/LM model respond to organic GDP growth when the money supply is held constant? If GDP increases (due to tech change/good investment), interest rates would move up along the LM curve and down along the IS curve, so what happens to the equilibrium? Do we assume GDP cannot grow without money supply growth, because an increase in interest rates along the LM curve would kill investment, bringing GDP back to equilibrium?

We would move upward along the LM curve. However if there is growth because of investment or some technology shock that would cause the IS curve to also shift upward.

IS: When the growth "shock" you mention happens, there is now more demand for goods (Demand= C+I+G in a closed economy), without any change in interest rates. Since demand shifts, supply goes along with it in IS-LM and GDP increases

The IS captures the GDP-to-interest rate relationship (along the curve) since this relationship has now changed, we must shift the curve. Specifically we have more GDP now per unit of interest rate compared to before. Since interest rates didn't change in your example, but there is more GDP (caused not by an interest rate increase) the IS curve must represent more GDP for each interest rate, which - if you draw it now - is a shift to the right of the IS.

LM: Represents what happens to interest rates when GDP increases. This relationship has not changed, so we don't have a shift. The reason this relationship has not changed is because the only reason interest rates are increasing now, is because of GDP, something the current LM curve already captures. (If it would be for something else, like money supply change, there would have to be a shift.)

So there can be growth without money supply increases. However since there is now more demand for money (because there are more things that we can buy) but not more supply of money, then money increases in value, i.e. the "price of money" which in a way is the interest rate must go up. For that reason, part of the growth will be reduced (as investment goes down), but we will still have growth. If money supply also increases exactly enough to accomodate the GDP increase (typically is the case in reality) the interest rate will not increase and the LM would also shift and we do not have the mentioned negative effect.


Two good sources (books preferably) that show the leading and most reasonable arguments for/against "trickle down economics"? I was raised by fairly conservative parents and throughout my childhood have had the impression that Reagan was the standard candle to which all policy should be held to and in particular Reaganomics was as true as the sky was blue. As an adult I've come to realize that it is not so cut and dry and would like some insight on this and would like to know if there are books that defend the for/against on the issue of "trickle down" economics. I would prefer to hear both sides in two separate books, or in a single book as long as both sides are fairly represented. Also please note I am not a student of economics although I have a strong math background.

This is a very large subject, where the terms have not even be well defined. I find it very unlikely that you will find an answer. Before even attempting to do so, you should define concisely what is meant by the terms. And particularly, are you talking about stabilization(short run) or structural(long run) economics? Because those two things are entirely different. One strategy is to study the classical and Keynesian schools of economics, since generally it at least to me appears that classical economics is called supply side economics, while keynsians are called demand siders.

Since the question was partly related to a person and not schools of economics, you could define "Reaganomics" as what actions he actually took, and what happened as a result. Upon doing this you will probably find that he wasn't really a strident supply sider in the first place. His strategy seemed to be to reduce taxes and pay for it by taking debt. Since it was not even a recession, this can be hardly classified as an economic policy, but a political strategy. Promise something for nothing and get votes is what seemed to be his plan in light of the data, and it worked as far as politics was concerned (he won twice).


Industries that thrive in crisis like COVID 19 I am aware that some industries thrive during crisis times like COVID 19...Food supplies, medical, counselling, funeral services, liquor etc.

Are there any other industries that may do well during these testing times?

In no particular order:

Consumer Staples: household goods, food, beverages, hygiene products, and other products that individuals are either unwilling or unable to eliminate from their budgets even in times of financial trouble
IT: Internet/Social Media, Software, and Semiconductors.
Healthcare: Anything that involves the provision and coordination of medical and related goods and services. Not to mention, small biotech companies.


Is it true that if marginal cost is constant, then average variable cost is also constant and equals marginal cost? I'm inclined to think yes because marginal cost only depends on variable cost (fixed costs don't matter), but I'm not 100% certain.
Basically, my thought process is that marginal cost of producing one additional unit is the change in total variable cost to produce that unit. So, my current idea is that if marginal cost is constant, that must mean that average variable cost (total variable cost/output) is also constant. But I'm still not entirely convinced.

Your intuition is correct.
First, you're right that "marginal cost only depends on variable cost", since
\begin{equation}
MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}.
\end{equation}
Next, if marginal cost is some constant $k$, then variable cost must be $VC(q)=kq$, because we can integrate $MC$ to obtain $TC$, where the term that varies with $q$ is $kq$:
\begin{equation}
MC(q)=k
\quad\Rightarrow\quad
\int k\;\mathrm dq =\underbrace{kq}_{VC(q)} +\underbrace{K\vphantom{q}}_{FC}
\end{equation}
where $K$ is a constant resulting from the integration.
Lastly,
\begin{equation}
AVC(q)=\frac{VC(q)}{q}=\frac{kq}{q}=k,
\end{equation}
which is a constant, and same as $MC$.


Pareto optimality with externalities I'm reading the book "Economics of natural ressources and the environment" written by D. Pearce and R. Turner. They explain that the competitive equilibrium is not a Pareto optimum in case of (positive or negative) externalities. The example they use is a firm that has an activity Q and this activity creates some pollution. They illustrate it in figure 4.1 (figure below). MNPB stands for "marginal net private benefits" and MEC "marginal external cost". MEC is thus the extra damage done by pollution arising from the activity measured by Q.

The firm will try to maximize its benefit, i.e. reach an activity of Q$^\pi$. Q$^*$ is the level of activity to have a social optimum. I understand the reasoning so far.

What I do not understand is why Q$^\pi$ is not Pareto optimum. If we had to reach Q$^*$ from Q$^\pi$, the firm would loose some of its utility so Q$^\pi$ should be Pareto optimal.

I feel like I do not understand the exact meaning behind the notion of the Pareto optimality.

Source of image: "Economics of natural ressources and the environment", D. Pearce and R. Turner, Harvester, New York, 1990.

I feel like I do not understand the exact meaning behind the notion of
  the Pareto optimality.

It's not you.  There are different senses of the phrase "Pareto Optimal," and you have to figure out from context which one is being used.  The dictionary definition of Pareto Optimal is something like "An allocation from which any feasible change which makes any agent better off makes at least one agent worse off."  The word to keep your eye on is "feasible."

In particular, it makes a big difference whether or not you consider side-payments feasible.  In your example, if it is impossible to make side payments from "society" (i.e. whomever is harmed externally by the production of the good) to the producer, then $Q^\pi$ is Pareto Optimal.  Of course, with no side payments, every point between $0$ and $Q^\pi$ is Pareto Optimal.

Almost always, however, we implicitly assume that side payments are possible.   In that case, the point $Q^\pi$ is not optimal.  Why not?  Well, we could move from $Q^\pi$ to $Q^*$.  This would make the producer worse off by $C$ and "society" better off by $C+D$.  Now, if it is feasible to make "society" pay the producer some amount between $C$ and $C+D$, then we can say that moving from $Q^\pi$ to $Q^*$ while compensating the producer at least $C$ at the expense of "society" makes everyone better off.  The fact that you can move off of $Q^\pi$ while making everyone better off means that $Q^\pi$ is not optimal.

This actually leads into a common criticism of Pareto Optimality.  Normally, we just assume side payments are feasible.  In fact, you get so habituated to making the assumption that it becomes invisible to you.  So, you start to think (for example) that "free trade is Pareto Optimal" is a good argument for free trade.  And that's untrue.  The necessary side-payments are probably not feasible and are not actually made in practice.


How to intuitively understand the 'Intuitive criterion'? The intuitive criterion by Cho and Kreps is a refinement to minimise the set of perfect Bayesian equilibria in signalling games. What would a simple and intuitive example to explain this criterion be? Assume any undergrad student should be easily able to appreciate the refinement through the example.

A concise, completely informal way of putting it is this: The intuitive criterion rules-out any out-of-equilibrium beliefs that can only be correct if some player did something stupid.
Below is a slightly more long-winded explanation with an informal example.

In many signalling games (that is, games in which one player—the sender—can communicate information to another—the receiver), there are often a lot of implausible equilibria. This happens because the Perfect Bayesian solution concept does not specify what the receiver's beliefs must be when the sender deviates; we can therefore support a lot of equilibria simply by saying that if the sender deviates from those equilibria then he will be "punished" with very bad beliefs. Such punishment will usually be enough to make the sender play a strategy that would otherwise not be a best response.
For example, in Spence's classic job market signalling paper there is an equilibrium in which high-ability individuals invest in education (learning is easy for them) whilst low-ability individuals do not (because they find it too costly to do so). Education is then a signal of ability. We might ask: is there also an equilibrium of this game in which nobody chooses to get an education and no information is transmitted to the receiver? The answer is 'yes'. We can support such an equilibrium by saying that a deviation in which a sender is educated causes the receiver to adopt the belief that the sender is certainly low-ability. If education has the effect of signalling low-ability then, of course, everyone is happy to play along with the putative equilibrium and not get educated.
It is also clear that this equilibrium is not very plausible: the receiver knows that it is less costly for a high-ability agent to get an education than a low-ability one, so it doesn't make much sense for him to think of an education as signalling low-ability. The intuitive criterion rules out this kind of equilibrium by requiring beliefs to be "reasonable" in the following sense:

Suppose the receiver observes a deviation from the equilibrium. The receiver should not believe that the sender is of type $t_{\text{bad}}$ if both of the following are true:

the deviation would result in
type $t_{\text{bad}}$ being worse off then if he has stuck to the equilibrium for any beliefs.
there is some type $t_{\text{good}}$ who is better off by playing the deviation than by sticking to the equilibrium for some belief other than $t_{\text{bad}}$.

Returning to the education signalling model: Suppose that the equilibrium is that nobody gets an education and that the receiver believes that a deviation to getting education signals low ability. Anticipating these beliefs, a low ability worker is made worse-off by deviating because he not only incurs the cost of the education but is then thought of as a bad type as a result. Thus, condition 1. is satisfied.
Can we find some alternative belief such that the high-ability worker would like to deviate to getting education? The answer is yes: if the receiver believes that education signals high ability then this deviation is indeed profitable for the high-type. Thus, condition 2 is also satisfied.
Since both conditions are satisfied, the intuitive criterion rules-out the implausible pooling equilibrium.


How does the professional class fit into Marx and Engel's class struggle theory? I know that in the first part of The Communist Manifesto, Marx and Engles discuss at length the relationship between the Bourgeois and the Proletarians. The former being the owners of the means of production and the latter being the working class that is engaged in a struggle against said owners.

However, after reading the first part of the manifesto I can't figure out where "Professionals such as doctors, layers, accountants, nurses, etc. fit into this model. While they aren't the owners of the means of production for physical goods they posses special knowledge that allows them to charge a premium for their services. As such, members of the professional class generally aren't viewed as struggling workers.

Why wasn't this class of worker included in the Communist Manifesto and where do they fit in with Marx and Engel's theory?

Marx's discussed struggle had less to do with your income-position in society and more to do with how you derive income, whether through profits, wages, or rents. You bring up a fascinating issue with his analysis and the solution is all about framing. Even doctors end up selling their labor on the market (in increasingly specialized components). While the labor of a doctor is different than say a machinist, their income is fundamentally different than that of someone who owns the means of production. The 17th century putting- out system would be maybe the clearest split between the two.  Keep reading Marx! Eric Fromm has some great essays that explain the nuance in Marx's world view. Hopefully this answer helps


Bayes-Nash equilibrium in Auction Consider a seller offering a single object for sale to two buyers with
  independent valuations – each bidder’s value is uniformly distributed
  on [0,1]. Assume that buyer 1 submits a bid that buyer 2 observes.
  Buyer 2 then has the ‘first refusal’ right to buy the object at that
  price. If she refuses, buyer 1 gets the object at the price he bid. What is the optimal strategy for buyer 1?

I have attempted the following:

$$\pi_1(v_1,v_2)=p(v_2<\beta(v_1))(v_1-\beta(v_1))$$
$$=(1-\beta(v_1))(v_1-\beta(v_1))$$

If this is an equilibrium, we must have that

$$=(1-\beta(v_1))(v_1-\beta(v_1)) \geq (1-\beta(w))(v_1-\beta(w))$$

for all $w\in[0,1]$. Hence,

$$\left.\frac{\partial \pi_1(v_1,v_2,w)}{\partial w}\right|_{w=v_1}=-v_1\beta'(v_1)-\beta'(v_1)+2\beta(v_1)\beta'(v_1)=0$$

$$\iff\beta(v_1)=\frac{1+v_1}{2}$$

But this strikes me as clearly wrong, if $v_1=0$ why should the player overbid? What am I missing?

I think your one mistake was writing $\Pr(v_{2} < \beta(v_{1})$ incorrectly. It should be:

$$\begin{split}
\pi_1(v_1,v_2)&=\Pr(v_2<\beta(v_1))(v_1-\beta(v_1))\\
&=F_{2}(v_2<\beta(v_1))(v_1-\beta(v_1))\\
&= \beta(v_1)(v_1-\beta(v_1))\\
\end{split}$$

$$\begin{split}
\left.\frac{\partial \pi_1(w,v_{2})}{\partial w}\right|_{w=v_1} =v_1\beta'(v_1) - 2\beta(v_1)\beta'(v_1)&=0\\
\beta(v_1) &= \frac{v_{1}}{2}\end{split}$$

Loosely we can look at, say, type $1$ of player $1$. If he bids according to this strategy his expected payoff is $1/4$. Suppose he picks any other bid amount $a$. Then his expected payoff is $(1-a)F(a) = (1-a)a$. This is clearly maximized at $a = 1/2$.


Why are we comparing the ratios of MPL to price of labor and MPK to price of capital not ratios of MPL and MPK to price of labor and price of capital Question:

A firm produces chairs using labor and capital. The price of labor is \$50 dollar per unit, and the price of capital is $100 per unit. At current output, the marginal product of labor is 10 chairs, and the marginal product of capital is 15 chairs. To reduce the total cost of producing the current quantity of chairs, how should the firm change its spending on labor and capital?

Answer:

Increase labor; Decrease capital. In order to reduce the total cost of producing the current quantity of chairs, one of the inputs to production must increase and the other must decrease. If neither changed, the cost would not be reduced. If one increased or decreased and the other did not change, the firm wouldn’t be producing the current quantity of chairs – it would be producing more or less, respectively. While the marginal product of labor (MPL = 10) is less than the marginal product of capital (MPK = 15), the RATIO of the MPL to the price of labor (10 / 50 = 0.2) is greater than the RATIO of the MPK to the price of capital (15 / 100 = 0.15). To reduce the total cost, the firm should increase spending on labor, which increases output by 0.2 chairs for every dollar spent, and decrease spending on capital, which only increases output by 0.15 chairs for every dollar spent.

You can do both. Effectively you say "let me move one dollar from labor to capital expenditure and see what happens". By doing so you have reduced labor by 0.01 and thus reduced production by 0.15 chairs, but increased capital by 0.02 and thus production by 0.2 chairs. Net effect is 0.05 chairs more at the same cost.

Equivalently, MPL/MPK = 1.5< w/r =2, or MPL/w= 0.15 < MPK/ r = 2. Second is just rearrangement of the first, but with both you answer where the dollar of you cost is more efficient.


Problem finding Demand Function I am analyzing pricing for a new streaming service for a school project. My thought is to calculate the demand function of a player in the same industry and then calculate the optimal price. I am a little confused about calculating the demand function for Netflix. I have their subscription fees (converted to annual) since 2007 - x axis and subscribers - y axis. I am getting an upward slope.

Below is the output of regression.

Subscribers = 76.5 + 0.5 Fee_per_year

I would appreciate it if someone can point out what mistake I am making.

You cannot empirically calculate the demand function in your current setting. Finding a demand function is indeed very difficult in general.

What we can observe, i.e. the data you have, is the market equilibrium over time, i.e. the equilibrium price and quantity levels. These are determined by the interplay of supply and demand. So when you observe a price and quantity you are not observing demand, but rather a combination of demand and supply.

To find the demand curve, you would need, for example, an instrument variable, i.e a variable that is correlated with demand, but independent of supply.


What is the impact of fracking on oil prices? I read an article about the recent fall of gas prices in the US claiming that the fall in gas prices was due to the expansion of fracking within the US.  The article said that, over the past few months, the oil harvested from fracking had come onto the market reducing the price of gas at road-vehicle refueling stations.

Obviously, I realize that an increase of supply with a constant demand will decrease prices.  But, is there any way to validate the article and conclude that the decrease in oil prices was due to fracking?  The article I read did not go into detail on how they reached their conclusion.

over the past few months, the oil harvested from fracking had come onto the market reducing the price of gas.

Obviously, this makes no sense insofar as the oil harvested from fracking displaces oil harvested by other means.

Absent changes in supply or demand, the price of oil has to rise at the rate of interest.

The development of a new extraction technology is a one-shot increase in supply, which should lead to a one-shot fall in price, followed by a resumption of the ongoing increase at the rate of interest.  It can't (by itself) explain an ongoing fall in price.


Is complex analysis used in economics? It's frequently useful in physics and engineering applications; are there any applications in theoretical economics? (If not, were there any attempts at incorporating CA that just never caught on?)

See http://en.wikipedia.org/wiki/Complex_analysis.

It should be pointed out that just because one encounters complex numbers does not mean one is doing "complex analysis", e.g. complex eigenvalues, complex Borel measures, Fourier transforms, etc. where trivial properties of complex numbers come up.

Complex analysis is a very focused subject unlike, say, real analysis, which is eclectic by comparison. At its core are holomorphic functions of one or more complex variables.

This paper

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693

is a specific instance of an economic model where complex analysis is used. The model solution technique used there is the identification between holomorphic functions on the unit disk and their continuation on the boundary. (The resulting function space is called the Hardy space, which contains the players' strategy spaces in the game being played in the paper.)


Do countries invest in the country's own stock market? For  example can the government of the UK invest in some company with shares in the FTSE 100? Or can Russia take the opportunity now that the shares of oil companies are low to buy them?

One example of monetary policy is the Bank of Japan buying the Nikkei as part of its asset purchasing program.

Typically, a substantial share of citizens' wealth is connected to returns of their national stock market.  Hence, it is generally not a wise move for a fiscal authority to invest public funds (not tied to pension schemes) in broad baskets of stocks.

You might try looking into sovereign wealth funds, these are typically gargantuan profit-maximizing investment vehicles wealthy nations use to acquire risky securities.


Oil prices forecast methods Oil prices have a significant impact on the world economy, positive for exporters and negative for importers to oversimplify a bit.
They are also known to be highly volatile, and depend on hard to predict parameters, such as the changing policy of the OPEC which resembles a cartel.

Most international organizations (European Commission and IMF for instance), seems to use future prices.

Do you have any studies mentionning or better menchmarking oil prices forecast methods ?

There is a lot of literature on this question.

This review looks like a good place to start although it is from 2013. It looks at a wide range of both qualitative and quantitative approaches.

  Among them, the most frequently used technique is the time series econometrics in order to forecast the
  volatility of oil prices, the second frequently used is the financial method, the third frequently
  used methods are based on structural models and non standard computational models, and the
  least used technique is the qualitative knowledge based method


McCall Search Model: Present value of lifetime income In the Bellman for the search model, how is the present discounted value of future income equal to $\frac{w}{1-\beta}$? Sounds like a basic question but I just can't comprehend the math in my head, because for present value I tend to compare it with NPV and think $1-\beta$ should be raised to the power of something. If someone could show the math in the McCall model's bellman.

In the version of the McCall search model I'm familiar with, once you accept an offer of wage $w$, you receive that wage $w$ in every period starting from the period of acceptance.

Hence, from the perspective of the current period, your utility from accepting that offer is

$$ w + \beta w + \beta^2 w + \cdots =\frac{w}{1-\beta} $$

In case this equality isn't familiar, recall that the LHS is simply a geometric series.


Model of economic cycles consistent with efficient market hypothesis I do not have background in economics, so my question may be naive or I might have taken incorrect assumptions somewhere. As far as I understand, efficient market hypothesis applies, saying that the stock prices already contain all the available information, so it is not possible to consistently beat the market without internal information on given stock.

I tried to build a simplistic toy-model of stock prices that have this property and I came with model, where each individual stock is in each time step multiplied by term $e^\varepsilon$, where $\varepsilon$ is randomly selected from uniform distribution centered around zero.

$$
p(t+dt)=p(t)e^{\varepsilon \,dt}\approx p(t)(1+\varepsilon \,dt)
$$

Now, since every positive $\varepsilon$ has equally probable negative counter-part and since $e^\varepsilon e^{-\varepsilon}=1$, the mean value of the stock price is zero. Unless I overlooked something, it should not be possible to extract profit from stock prices following such a law, because the information about the future price is not contained in the past and it is random. We can center the distribution around positive number to add growth of the overall economy.

Now I wanted to create similar toy-model, where economic cycles are included. First thing that came to my mind is to multiply the generated prices of stocks from the previous model by a periodic function, for example:

$$
p_\mathrm{cycled}(t)=\left(1+A\sin\left(\frac{2\pi t}{T}\right)\right)p(t)
$$

But this is clearly contradicting the efficient market hypothesis, because anyone knowing the period of the function could buy the stock when the prices are low and sell when the prices are high. I thought of possible ways how to make the model consistent with the efficient market hypothesis and I came with these points:

The cycle is not a cycle, but some kind of noise. What makes the strategy of making profit buying low and selling high impossible is, that nobody knows, when the prices will go up and when down.
When buying low, some companies go bankrupt before going up to high period of the cycle. Nobody knows which one though, and it is therefore not possible to make profit on the cycle.
Something else.

The question: Is there simple toy-model of stock prices that manifests the economic cycles and is consistent with the efficient market hypothesis? If there are more mechanisms at play, I would be very interested in seeing them all embedded as a toy model.

I am suspicious about 1., because either the noise is unbounded (as the exponential noise above), or there is memory in the noise. The memory can be used to extract the necessary information to make profit. If the noise is unbounded, then it is not really similar to how net world economy behaves. 2. seems plausible, but I didn't found any toy-model embedding such behavior yet.

-----------------------------

My takeaway:
From the answers here and after giving it some thought, I came to conclusion that if the EMH is true, there should not be any cycles with known oscillations in the prices, no autocorrelation in the noise to be exploited and the possible cycle up- and down- should not be bounded (otherwise, everyone would know when is the down-phase, even though they would not be able to predict it).

I came with example toy-model satisfying these requirements with three components:

Normal distribution random walk
Exponential growth of several % per year
Constant small probability of sudden crash with factor 0.75-1

In the model, there are "quasi-cycles" between the crashes and there is no exploitable future information. (I think.)

The EMH applies to assets, not just stocks, and its implications are more relevant for investors who own part of the market - not the entire thing.  This is important, because it's the difference between looking at a closed system versus an open one, and between populations versus samples.  People make money all the time by cycling between stocks, bonds, precious metals, real estate, etc.  Flights to "safe havens" occur all the time, rightly and wrongly.  Someone buying SNC Lavalin stock three months ago and selling it a few weeks after the Canadian election would have made some fairly predictable money.

The apparent issue you're running into here stems from the erroneous assumption that non-stock assets are "safe".  In fact, they have their own risks and cycles, which are (anti-)correlated with the stock market in various ways.  The "Risk-free rate" in the Fama-French model isn't constant, and is fundamentally no more predictable than any individual stock price.


Question Regarding Equilibrium Price & Surplus So here's an example of a standard Supply & Demand Relationship for an Individual Supplier:

As a supplier, for 1 dollar, I'll produce 1 unit of something. For 2 dollars, I'll produce 2 units, for 3 dollars, I'll produce 3 units, and so on and so forth up until 10 dollars for 10 units.

At 1 dollar, I'll have 10 buyers, at 2 dollars, I'll have 9 buyers, at 3 dollars, I'll have 8 buyers...all the way up to having 1 buyer at 10 dollars.

As you can see, there's an inverse relationship with the equilibrium price being right in the middle: Producing 5 units with 5 buyers buying each of them at 5 dollars per unit. Simple...

But now I'm wondering what would happen if there was a surplus. Ideally I should've made 5 units at 5 dollars each and sold them all to 5 buyers. But let's say I make 6 units, thinking I'll be able to sell all of them for 6 dollars each, but I only end up selling 4 of them to 4 buyers. If I wanted to sell all 6, I would have to sell them at 4 dollars each to reach equilibrium. The problem is if I sell 6 of them for 4 dollars each, I'll make less money than I would selling only 5 of them for 5 dollars each.

So do I reduce the price to 4 dollars so I can sell all 6 of them, making 24 dollars? Or do I sell 5 of them at 5 dollars each, making 25 dollars, and then sell the remaining one for less money?

Thanks in advance for answers.

A supply curve is typically derived from a cost function, and you're leaving out some important costs such as storage and depreciation, which would help relieve some of the seeming counter-intuitiveness around a supplier choosing to "sell more at a loss" compared to selling less.


How to calculate equilibrium income given a certain level of unemployment? I need help with this question. I do know the procedure to calculate the equilibrium income and can solve the following question without taking the employment factors in mind. However, I just can't get how the employment factor figures in the equilibrium calculations.

Suppose labour and capital are used in fixed proportions in the production of output in a closed economy. Suppose the existing capital equipment is utilised fully but a quarter of the labor force is unemployed. Let $Y$,$C$ and $I$ represent aggregate levels of income, consumption, and investment expenditure in the economy in dollars at constant prices. Let $C=100+0.6Y, I=500, G=200$

Calculate the level of income in the economy.
If the government raises its nominal expenditure to 400, then what is the increase in real income in the economy?
If the government raises its nominal expenditure to 400, then what is the fall in the percentage of unemployed in the economy?

This is how I've solved it :
Y=C+I+G = 800+0.6 Y So Y = 2000
With 400 as government expenditure, the new equilibrium income will be 2500. So the increase in the income is 500.
I think this will be solved using Okun's law:

$$\Delta u_{t}=0.4(g_{y_{t}}-3)$$

But I'm not sure how to proceed.

this is how I've solved it :

(a) Y=C+I+G = 800+0.6 Y
    So Y = 2000

(b) With 400 as government expenditure, the new equilibrium income will be 2500. So the increase in the income is 500.

(c) I think this will be solved using Okun's law ut-ut-1=0.4(gyt-3%). But I'm not sure how to proceed.


Why do interest rates move together? Why do interest rates (mortgage, LIBOR, Treasury yields, etc.) move together? What is the fundamental reason behind that?

From my understanding, what drives it is the Federal Funds Rate, which is the rate at which banks borrow from the Federal Reserve. If that rates decrease, then they can borrow at a lower rate, which in turn impact their loan business because they can issue lower rate loans.

Are there other reasons why that is the case? And what are the cases where rates do not move together?

The assumption is based on intuition but it a bit more complicated:
Basically each and every country has short term interest rates and long term interest rates, based on the debt the country is managing.

Short term interest is usually being set by central banks (the fed in the US), when the long term debt is managed by treasury. long term affect mortgages while short terms affect, short term loans and deposits
Long and short term rates affect each other - the long term interest, being riskier, in the same country, will yield more

In today's open global capital markets, money seeks the best returns. with free money flows, so the best interest - inflation ( - risk) will pull more capital
And on the other hand, it will increase the country's currency value (higher demands for the currency). This in turn, will affect exporting, for each country.
Since this game is a global game between countries, they will pretty much react to each other's actions, effectively matching rates and policies

Note that the description I gave may sound weird, and is a bit contradictory of economic theory, but that is the reality for the past decade...

Monetary policy in the world today, is in a theoretical no mans land


Does the inflation rate reflect only the change in quantity of money? Does inflation rate during the years comes from the the proportional difference of current amount of currency and amount of currency 1 year ago.

I mean lets suppose in 2000 there were 1 million dollars in flow and in 2001 100k were added into the flow, so the inflation rate is 11%?

The growth of the amount of money in the economy is just one factor that can affect inflation. Other factors include:

changes in a nations productivity
changes in the exchange rate with other countries
changes in the velocity of money
changes in what fraction of financial exchanges are counted in inflation incidences

Given all these additional factors, its no surprise to find that in the short term the money supply does not correlate brilliantly with inflation... but in the long term the correlation is definitely there - i.e. if the money supply grows by a factor of ten over many years, then prices would probably grow by a similar factor. Milton Friedman won his Nobel prize for his work highlighting the long term correlation.


Why does capital initially reduce and then rise with an anticipated future decrease in taxes in the RCK model? Consider the version of the RCK model where there is a government that runs a constant balanced budget.

  At $t_0$, the economy is in steady state, with constant tax $T_{old} > 0$
  Then, at $t_1$, it is announced that at some later moment in time ($t_2$), but not right now, tax will be permanently reduced to $T_{new} < T_{old}$.

  Draw a graph of changes in the phase diagram that characterizes the dynamics of the economy, and heuristic graphs of the time paths of the following variables:

  consumption per effective worker
  capital per effective worker
  marginal return on capital per effective worker ($r_t$)

The memo gave the following:

Why do the time paths of capital and the real interest rate look like that?

When the economy transitions between time $t_1$ and $t_2$ (the red line in the phase diagram), capital reduces (because of the leftwards movement). That is why the green line showing the path of capital over time goes down initially. Because capital reduces, the marginal product of capital increases (as seen by $r$ increasing in the time path between $t_1$ and $t_2$.

Then, the economy is on the saddle path, and gradually returns to the new balanced growth path, which is a point with the permanently lower tax rate (and thus permanently higher consumption spending). When this gradual change happens (shown as the black arrow in the phase diagram), capital increases (movement to the right), so in the time path, the green line goes back up to the initial level. Because capital returns to its initial level, the return on capital also decreases to its initial level.

Consumption initially jumps up because disposable income over time is discounted, according to the formula $$\int_{t=t_1}^\infty e^{-R(t)}(W(t) - T(t)) \mathrm dt $$ where $R$ is the discount factor, $W$ is the wage and $T$ is tax, so even though disposable income doesn't decrease immediately, the anticipated decrease in the future affects current consumption. Other than that, the anticipated permanent change in government spending puts consumers on the new saddle path.


Does Chile have a government owned Export Import bank like the US or India? If not, what type of trade financing do Chilean exporters or importers get? Is it all from private banks and insurers?

Chile joined the OECD in 2010 but does not have an official export credit agency listed by the OECD in its involved in the Export Credit Group

The Inter-American Development Bank is lending US$27 million to the Chilean Finance Ministry to support export promotion and services, but this does not look like financing

So I would assume that financing comes from private sources (in particular Chile has large private pension systems) and foreign export credit agencies, plus the usual market operations of the Chilean central bank to ensure monetary stability and liquidity in the context of a floating exchange rate significantly affected by the world price of copper


An approximate measure of the real interest rate applying rule of thumb for growth rates I don't understand how one can get from:

$1+r_{t+1}$=$\frac{1+i_{t}}{1+\pi_{t+1}}$

to

$1+r_{t+1}$$\approx1+i_{t}-\pi_{t+1}$

by applying the following rule of thumb:

$Z$=$\frac{X\cdot{Y}}{Q}$

implies

$\frac{\Delta{Z}}{Z}$=$\frac{\Delta{X}}{X}$+$\frac{\Delta{Y}}{Y}$-$\frac{\Delta{Q}}{Q}$

Note:

$r_{t+1}$: real interest rate

$i_{t}$: nominal interest rate

$\pi_{t+1}$: inflation rate

I hope someone will help :)

Argument 1:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$

Take the natural log of both sides.

$$ \log(1+r_{t+1}) = \log(1+i_{t}) - \log(1+\pi_{t+1})$$

Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the first order taylor expansion. Hence:

$$ r_{t+1} \approx i_{t} - \pi_{t+1}$$

Argument 2:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$
$$  1+r_{t+1} + \pi_{t+1} + r_{t+1} \pi_{t+1} = 1+i_{t}$$

And if $r_{t+1} \pi_{t+1}$ is closer to zero

$$ r_{t+1} + \pi_{t+1} \approx i_{t}$$


Finitely repeated prisoner's dilemma without sub-game perfection Suppose that two individuals play the prisoner's dilemma (PD) a finite number of times; and assume that they both discount the future at a constant rate. Can cooperation be sustained by a Nash equilibrium? Notice that I am not restricting attention to sub-game perfect Nash equilibria (obviously, there are no SPNE which sustain cooperation).

There is also no NE which sustains coopration for more or less the same reason as in the SPNE case.
Consider, a PD played twice. A strategy contains five actions, one for each decision node: one in the beginning (empty history) and one for each of the four period-2 histories (CC,CD,DC,DD). I claim that any strategy other than (D;D;D;D;D) is dominated.
Consider any strategy in which you play C in period 2, say (C,D,C,C,D). A deviation to (C,D,D,D,D) is profitable because defection cannot be punished after period 2 as the game ends. Behavior in period 1 cannot be conditioned on the future. Given that any equilibrium candidate has the structure (_,D,D,D,D) cooperation in period 1 is also dominated.
If you are not convinced, you can write down the game with all its strategies in a big normal form matrix.
You can iterate the argument for any (commonly known) finite number of periods.


Why do many countries hoard gold bars? In my country, the only legal way to sell a gold bar is to sell it to the central bank. What are the reasons why some countries hoard gold bars?

The answer from the former Chair of the Federal Reserve is in a word: tradition.

Chair Ben Bernanke testifying to the United States Senate.


Can a game with a unique pure strategy Nash equilibrium also have a mixed strategy equilibria? Consider an arbitrary 2x2 simultaneous game with complete information. Say that the model has only one pure-strategy Nash equilibrium. For example (first pay-off refers to Player 1):

                    Player 2
           +---+-------+-------+
           |   |   A   |   B   |
           +---+-------+-------+
Player 1   | A | (1,2) | (2,1) |
           | B | (3,3) | (1,1) |
           +---+-------+-------+

Here, (3,3) is the only pure-strategy Nash equilibrium. The reason there is just one is, apparently, because one of the players have a dominant strategy (Player 2 always prefers A). If we change this, we would get two pure-strategy equilibria.

Abstracting from the fact that the pure-strategy equilibrium is also a mixed strategy one with probability 0/100%, does any simple game with just one pure-strategy equilibrium have no mixed strategy equilibrium? Is there a formal proof of this?

I educated guess is that there is no mixed strategy equilibrium. The "proof" I can think of is a best response plot. Basically, one player has a dominant strategy which means that for any probability, her response is a vertical/horizontal line at 0 or 1 (depending how probability is defined). Thus, regardless of how the other player's best response line looks like (i.e. where the indifference probability is), it will only cross the other player's line in one point. That point is the pure-strategy equilibrium.

PS: the adjective "simple" is to avoid more complex game scenarios like repetition, cooperation, incomplete information, etc.


Question about simple SRAS and AD curve Does the general price level show the aggregate of all goods & services demanded or supplied at a certain price or does it show something else entirely?

My knowledge of economics is still very much in its infancy, so only plain language in answers please!

General price level is defined by the price that you pay for a basket of goods. This basket of goods is decided on the basis of general consumption patterns in the economy and includes a number of sectors (with a number of items) like food, transport, medical care, etc. Such baskets can be defined differently to arrive at different measures of general price levels such as the one faced by end consumers or wholesale buyers.

In a nutshell, it gives an idea of how the prices in the economy are moving. So, price of a good A might increase while price of good B might decrease but it may so happen that the general price level may increase or decrease depending upon the defined basket and the weights assigned to the items in basket.


Simulating Real Business Cycle Basically I need to replicate Hartley's 'A User's Guide to Solving Real Business Cycle Models' (http://www.econ.ucdavis.edu/faculty/kdsalyer/LECTURES/Ecn235a/Linearization/ugfinal.pdf). Specifically, I want to simulate the dynamical system implied by the model which is specified as follows:

where $c$ is consumption, $h$ is labour supply, $k$ is capital, $z$ is the autoregressive technological process, $y$ is the output and $i$ is investment.

I simulate it using the following logic: say at time $t$, everything is at steady state and all the values are 0, from which we have $k_{t+1}$. Then, at $t+1$ by giving a shock to the system through $\varepsilon$, i solve for $c_{t+1}$ and $h_{t+1}$ (as I have the 'shocked' $z_{t+1}$ and previously obtained $k_{t+1}$. Then, I plug those two to retrieve the rest, namely - $y_{t+1}, i_{t+1}, k_{t+2}$ and repeat the process.

Unfortunately, I get an explosive process which doesn't make sense:

I also include R code that is used to simulate this:

n<-300

data.simulated <- data.table(t = 0, zval = 0, cval = 0, hval = 0, kval = 0, yval = 0, ival = 0)
data.simulated <- rbind(data.simulated, data.table(t = 1, kval = 0), fill = TRUE)

for (ii in 1:n){

  ##initial shocks
  eps <- rnorm(1, mean = 0, sd = 0.007)
  zt1 <- data.simulated[t == ii - 1, zval]*0.95 + eps
  kt1 <- data.simulated[t == ii, kval]

  ##solve for ct, ht
  lmat <- matrix(c(1, -0.54, 2.78, 1), byrow = T, ncol = 2)
  rmat <- matrix(c(0.02 * kt1 + 0.44 * zt1, kt1 + 2.78 * zt1), ncol = 1)

  solution <- solve(lmat, rmat)
  ct1 <- solution[1, ]
  ht1 <- solution[2, ]

  ##now solve for yt1 and kt2 and it1
  yt1 <- zt1 + 0.36 * kt1 + 0.64 * ht1
  kt2 <- -0.07 * ct1 + 1.01 * kt1 + 0.06 * ht1 + 0.1 * zt1
  it1 <- 3.92 * yt1 - 2.92 * ct1

  ##add to the data.table the results
  data.simulated[t == ii, c("zval", "cval", "hval", "yval", "ival") := list(zt1, ct1, ht1, yt1, it1)]
  data.simulated <- rbind(data.simulated, data.table(t = ii + 1, kval = kt2), fill = TRUE)
}

a <- data.simulated[, list(t, cval, ival, yval)]
a <- data.table:::melt.data.table(a, id.vars = "t")
ggplot(data = a, aes(x = t, y = value, col = variable)) + geom_line()

Sy my question is simple - is the system specified in the paper is inherently unstable and ergo the results, or did I make a mistake somewhere?

Explosiveness

The paper contains an error, which causes the explosive dynamics in your simulation (although presumably the underlying computations in the paper were correct). The equilibrium condition derived from eigenvalue decomposition is contained in the third row of matrix $Q^{-1}$ on page 12 of the paper, with variables ordered as $(c,k,h,z)$ (I'll drop tildas, so all lowercase variables are to be understood as log-deviations). Comparing with eqn. (16) on p. 13, we see that coefficients for $k$ and $h$ are switched, and so the correct condition is

$$
c_t = 0.54 k_t + 0.02 h_t + 0.44 z_t
$$

Simulation

First, we can express consumption and labor as linear function of state variables (no need to solve the system at each step of the simulation). The intertemporal and intratemporal equilibrium conditions can be written as

$$
\begin{bmatrix}1 & -0.02 \\ 2.78 & 1 \end{bmatrix} \begin{bmatrix} c_t \\ h_t\end{bmatrix} = \begin{bmatrix} 0.54 & 0.44 \\ 1 & 2.78 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix}
$$

so after multiplying by an inverse we get

$$
\begin{bmatrix} c_t \\ h_t\end{bmatrix} = \begin{bmatrix} 0.53 & 0.47 \\
-0.47 & 1.47 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix}
$$

Next, transition for states can be written as

$$
\begin{bmatrix} k_{t+1} \\ z_{t+1} \end{bmatrix} = \begin{bmatrix} -0.07 & 0.06 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} c_t \\ h_t\end{bmatrix} + \begin{bmatrix} 1.01 & 0.1 \\ 0 & 0.95 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix} + \begin{bmatrix} 0 \\ \epsilon_{t+1}\end{bmatrix}
$$

which can be reduced by substuting for control variables to

$$
\begin{bmatrix} k_{t+1} \\ z_{t+1} \end{bmatrix} = \begin{bmatrix} 0.94 & 0.16 \\ 0 & 0.95 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix} + \begin{bmatrix} 0 \\ \epsilon_{t+1}\end{bmatrix}
$$

Now the simulation should be trivial, here's a Matlab/Octave example:

T = 200;
X = zeros(2,T);
for i=2:T
    X(:,i) = [0.94 0.16; 0 0.95] * X(:,i-1) + [0; 0.007*randn()];
end
Y = [0.53 0.47; -0.47 1.47] * X;
figure
plot(1:T, [X; Y])
legend('k','z','c','h')

Of course in practice, you should probably recompute the whole solution, including the eigenvalue decomposition, so that you would be able to change parameters, etc.


How do I run a regression with the restriction that one set of parameters are proportional to another? I'm running a regression with a set of 3 dummy variables (for four categories of a variable) and these 3 dummies interacted with a continuous variable. I want to impose the restriction that the vector of the coefficients of the interacted dummies is proportional to the coefficients of the dummies, e.g. each interaction term has a coefficient 2x larger than the coefficient on it's non-interacted dummy.

Is it possible to impose these restrictions with a linear regression model? or will I have to use non-linear least squares?

(P.S. the proportionality is a ratio to be estimated, so that would be another free parameter in the model)

Just to make things simple I'll show how to do it for a simpler case.

Let's say you have a regression $Y = \beta_1 X_1 + \beta_2 X_2 + \varepsilon$ and you want to impose $\beta_1 = 2\beta_2$.

This is equivalent to saying $Y = 2 \beta_2 X_1 + \beta_2 X_2 + \varepsilon = \beta_2 (2 X_1 + X_2) + \varepsilon$. That is, you can generate a new variable $X_3 = 2X_1 + X_2$ and regress $Y$ on $X_3$.


How to deal with Prescott's formulation of time to build in his original RBC model? So I was replicating the results obtained in section 4 of Prescott's original paper, which derives optimality conditions in steady state without shock. I hope to solve the social planner's maximization problem by using the method of Lagrange.

An important feature in Prescott's model is that capital is built in multiple periods and each project costs resources, i.e., a fraction of output, in each intermediate period. My question is that how to formulate this constraint into the Lagrangian so I can solve the system of equations of FOC? Here is what I have now:
\begin{multline}
 \mathcal{L}=\sum_{t=0}^\infty\beta^tu(c_t,\alpha(L)l_t) -\sum_{t=0}^\infty\lambda_{1t} \left( k_{t+1} - (1-\delta)k_t-s_{1t}\right)
 -\sum_{t=0}^\infty\lambda_{2t}\left(i_t-\sum_{j=1}^J \varphi_j s_{jt}+y_{t+1}-y_{t}\right) \\
 -\sum_{t=0}^\infty\lambda_{3t}\left(c_t+i_t- f(\lambda_t,k_t,n_t,y_t)\right)
 -\sum_{t=0}^\infty\sum_{j=1}^{J-1}\left[\lambda_{j+3,t}\left(\sum_{t=0}^\infty s_{j,t+1}-s_{j+1,t}\right)\right]\\
\end{multline}
where $s_{j,t}$ is the investment project that takes $j$ more periods to complete as of time $t$ and $y_t$ is the stock of inventory at time $t$. For the purpose of this question, let's ignore the lag polynomial $\alpha(L)$ and assume it to be have effect on $l_t$. Note that $l_t$ is leisure and $1-l_t=n_t$ is labor input where time is normalized to 1. In addition, output goes to inventory, consumption, and capital projects. I have tried to use the following substitutions:
\begin{align}
 k_{t+1} &= (1-\delta)k_t + s_{J,t-(J-1)}\\
 i_t&=\left(\sum_{j=1}^J \varphi_j s_{J,t-(J-j)}\right)+y_{t+1}-y_{t}
\end{align}
which allows me to get rid to the last term in $\mathcal{L}$. However, I still do not seem to be making any progress. It will be great if you can give me suggestions on formulation of my lagrangian and what FOCs will give me the steady state consumption, capital stock.

EDIT: I tried many online lecture notes, but none of them seem to want to deal with Prescott's time to build formulation and alike...

By re-indexing all investment projects in the form of $s_{J,t}$, I get the following two equivalent constraints
\begin{align}
 c_t+y_{t+1}-y_t&\leq f(k_t)-\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)} % Resource Constraint
 \\
 k_{t+1}&=(1-\delta)k_t+S_{J,t-(J-1)} %Law of motion for capital formation
\end{align}
Hence, the Lagrangian for the maximization problem is
\begin{multline}
\mathcal{L}
 = \sum_{t=0}^\infty\beta^tu\left(c_t,1- \alpha_0n_t - \eta(1-\alpha_0) a_t\right) %Objective Function
 - \sum_{t=0}^\infty\lambda_{1t}\left(c_t+y_{t+1}-y_t+\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)}-f(k_t)\right) % Resource Constraint
\\ - \sum_{t=0}^\infty\lambda_{2t}\left(k_{t+1}-(1-\delta)k_t-S_{J,t-(J-1)}\right) %Law of motion for capital formation
 - \sum_{t=0}^\infty\lambda_{3t} \left(a_{t+1}-(1-\eta)a_t-n_{t}\right) % Law of motion for past leisure
\end{multline}
where the state variables at time $t$ are $a_t, k_t, y_t$ and the decision variables at time $t$ are $c_t, n_t, s_{Jt}, y_{t+1}$. It is easy to see that all constraints must bind.
The necessary conditions are
\begin{align}
 &\frac{\partial \mathcal{L}}{\partial c_t} =0\implies \beta^t \frac{\partial u}{\partial c_t}=\lambda_{1t}
\\ &\frac{\partial \mathcal{L}}{\partial s_{J,t}} =0\implies \sum_{i=1}^J\lambda_{1,t+J-i}\varphi_i=\lambda_{2,t+J-1}
\\ &\frac{\partial \mathcal{L}}{\partial k_t}=0\implies \lambda_{1t}\frac{\partial f}{\partial k_t}+\lambda_{2t}(1-\delta)=\lambda_{2,t-1},
\end{align}
which holds for all $t$.
Combining 14,15,16 and simplifying, we have
\begin{equation}
 \frac{\partial u}{\partial c_t}\frac{\partial f}{\partial k_t}
 + (1-\delta)\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t+1-i}}\varphi_i
 = \beta^{-1}\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t-i}}\varphi_i
\end{equation}
In steady state, $\frac{\partial u}{\partial c_t}=\frac{\partial u}{\partial c_{t'}}$, so we have
\begin{equation}
\frac{\partial f}{\partial k_t}
= \sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} \left(\frac{1-\beta}{\beta}+\delta\right)
\end{equation}
One observation is that since in steady state, no borrowing can occur between household and firm, the real interest rate $r=\frac{1-\beta}{\beta}$. We see that $\sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} $ is the price of the capital in steady state.


Expected Utility with expected value and variance I'm having trouble with a question from Ariel Rubinstein's book, Lecture Notes in Microeconomic Theory. It's the problem 2 from Problem Set 7. Here's the question:

  Show that the utility function $u(L) = \mathbb E(L) - (\mathbb E(L))^2 - var(L)$ is consistent with vNM assumptions.

Where $\mathbb E(L)$ and $var(L)$ are the expected value and the variance of the lotteries, respectively.

So here's what I thought: we know the following set of implications
$$\text{Function is linear} \implies \text{Has the expected utility form} \implies \succsim \text{satisfies vNM assumptions.}$$

So, it suffices to show that $u(L)$ is linear. As we know that $var(L) = \mathbb E(L^2) - (\mathbb E(L))^2$, let's rewrite our utility function.

$$u(L) = \mathbb E(L) - (\mathbb E(L))^2 - (\mathbb E(L^2) - (\mathbb E(L))^2) = \mathbb E(L) - \mathbb E(L^2).$$

Take two lotteries, $L$, $M$. We should have that

$$U(\alpha L + (1 - \alpha)M) = \alpha U(L) + (1-\alpha ) U(M) \qquad \alpha \in [0,1].$$

So, $$U(\alpha L + (1 - \alpha)M) = \mathbb E(\alpha L + (1 - \alpha)M) - \mathbb E((\alpha L + (1 - \alpha)M)^2) = \alpha \mathbb E(L) + (1-\alpha)\mathbb E(M) - \alpha^2 \mathbb E(L^2) - 2\alpha (1 - \alpha)\mathbb E(LM) - (1 - \alpha)^2 \mathbb E(M^2).$$

But the above isn't equal to $$\alpha U(L) + (1-\alpha ) U(M) = \alpha (\mathbb E(L) - (\mathbb E(L))^2) + (1 - \alpha)(\mathbb E(M) - (\mathbb E(M))^2).$$

Can you guys help me to see where I got it wrong?

It would suffice to show that $U$ is linear. But is $U$ necessarily linear if it satisfies the vNM axioms?

Hint: No.


LS book (Recursive Macro Theory). Chapter8: Complete Market I am reading LS book, on Chapter 8 - Complete Markets (3rd version). There is an example on page 264 that I am quite confused.

I attach the picture of the example here under.

I don't understand why we have a history (0,0,0,...,0,1,1)? since if $s_t=0$, then $s_{t+1} = 0$ for sure, right?

Or the authors means the history here is in order $(s_t, s_{t-1},....,s_1,s_0)$,
so it makes sense since we know that $s_0=1, s_1=1$, but isn't it quite weird, since we usually write a history $h_t=(s_0,s_1,...s_t)$ right?

Anyone has an idea? Really appreciate your help!

Since this is a Markov chain

\begin{eqnarray}
\pi_t(0,0,\cdots,1,1) &=&  \color{blue}{\pi(s_t = 0 | s_{t - 1} = 0)} \cdots \color{magenta}{\pi(s_2 = 0 | s_{1} = 1)} \color{red}{\pi(s_1 = 1 | s_{0} = 1)}\color{orange}{\pi(s_0 = 1)} \\
&=& \color{blue}{1} \times \cdots\times \color{magenta}{0.5}\times\color{red}{1}\times \color{orange}{1} = 0.5
\end{eqnarray}

But you are right if $t > 2$ then $\pi(s_{t + 1}  = 0 | s_{t} = 0) = 1$, so if $s_t = 0$ then $s_{t + 1} = 0$, but at $t = 2$ there's a 50/50 chance that the state changes from $1$ to $0$, so the state may remain $1$ as in the first history, or switch to $0$ as the second one


Having a lower inflation rate is not the same as having lower overall prices? Having a lower inflation rate is not the same as having lower overall prices.Why?

lower CPI(inflation) means price levels in country decreases?

The inflation rate is the rate of change in the price level.

A lower - but positive - inflation rate (say from 3% to 2%) still results in higher prices -this is typically called disinflation, and highlights the fact that lower inflation doesn’t necessarily mean lower prices.

A negative inflation rate print (like -1%) implies the price level did actually decrease - typically referred to as deflation.


Who first formulated the Political Business Cycle Theory? The first time I saw it, was in Stephen Harper's Master's Thesis

http://dspace.ucalgary.ca/bitstream/1880/24345/1/1991_Harper.pdf

It comes across as being very similar to the Austrian Business Cycle Theory, but I have not come across it in their writings as of yet.

Who was the one to first coin and formulate the theory?

Quick Wikipedia found :

https://en.wikipedia.org/wiki/Business_cycle

Look at the section of Politically based business cycle

  The political business cycle theory is strongly linked to the name of
  Michał Kalecki who discussed "the reluctance of the 'captains of
  industry' to accept government intervention in the matter of
  employment."

https://en.wikipedia.org/wiki/Micha%C5%82_Kalecki


Can you take the moving average of quarterly data of an explanatory variable in a regression to smoothen noise and get more accurate coefficients? I'm trying to use acceleration of quarterly data on household debt (the difference in the difference in debt) in a regression on unemployment (only concerned with correlation) but quarterly data is much noisier than annual data, which may weaken the correlation. I was wondering if transforming the already differenced (twice) data as a moving average would change my coefficients at all and if they would be more accurate. Is this a common technique in econometrics?

Let's say $D_t$ is the stock of debt at time $t$.

The first difference is $D_t - D_{t-1}$.

The second difference is $(D_t - D_{t-1}) - (D_{t-1}-D_{t-2}) = D_t - 2D_{t-1} + D_{t-2}$.

The [4-period] moving average is then given by

\begin{align*}
&= \frac 1 4 (D_t-2D_{t-1}+D_{t-2}+D_{t-1}-2D_{t-2}+D_{t-3}\\
&\qquad+D_{t-2}-2D_{t-3}+D_{t-4}+D_{t-3}-2D_{t-4}+D_{t-5})\\
&= \frac 1 4 (D_t - D_{t-1} - D_{t-4} + D_{t-5})\\
&= \frac 1 4 ((D_t - D_{t-4}) - (D_{t-1} - D_{t-5}))
\end{align*}

Which would be the yearly increase in the stock of debt this period compared to the same thing a quarter ago. Only you could tell if this is what you want in your regression. Something to consider is only to use the first difference: $D_t - D_{t-4}$.


Banking System, Canada & US Why are the way banks are established and spread out in Canada and US so different?

Canada:
There are only six major banks and most products offered are very competitive. Its almost unusual to hear or see any other minor or regional bank.

US:
Literally, there are thousands of banks. Although there are few big ones, like BoA, Wells Fargo, Chase, there are thousands of regional banks, even down to county levels.

What are the advantages of having multiple individual banks?

Why did US encourage this type of proliferation of banks while it did not happen in Canada?

Limiting the scope to the question of why the US encouraged the proliferation of many banks, one important contributor was the 1927 McFadden act, which placed significant restrictions on interstate banking (rolled back through state laws in the 1980s-90s, and eventually substantially repealed by the Riegle-Neal Interstate Banking and Branching Efficiency Act of 1994).

The reasons for this were primarily concerns about concentration in the financial services sector, and concerns about the ability of state banking supervisors to monitor across state lines the activities of institutions for which they were responsible.

The trend in the US since at least the 1980s has indeed been towards consolidation: http://research.stlouisfed.org/fred2/series/FREQ

Finally a (dated, but good) review of the effects of branch banking reform is available from the FRBNY: http://www.newyorkfed.org/research/current_issues/ci1-2.pdf


How can power/electricity prices be negative? Bloomberg shows this chart:

I understand the above chart to mean that consumers were actually paid to use electricity. (Please correct me if I'm mistaken.)

I was wondering how this is possible? How is the power/electricity market special and different from other markets? (Or perhaps the German electricity market is special.)

Yes, customers who were exposed to day-ahead wholesale market prices were paid to use electricity.

It is a combination of several different factors that make the day-ahead wholesale electricity market special.

Firstly, very few electricity consumers participate in it directly. So the demand side is very illiquid, with very low short-run elasticity - and it quickly becomes perfectly inelastic.

Secondly, there's very little time-arbitrage. The market for electricity at 0500 and the market at 1500 are two separate markets: there are very few arbitrageurs who can buy at 0500 and sell at 1500. That's because until recently, there's been very low rewards for doing so (more on this, below).

Thirdly, there are several suppliers for whom reducing supply would incur expense: so, for market-clearing purposes, they have negative short-run marginal costs. There are two ways that can happen.

very inflexible conventional generators (usually nuclear) just can't turn their output up and down every hour or two, without incurring maintenance costs.
generators with contracts that enable them to sell all their output at a pre-specified positive price: they would need wholesale prices to go negative by the more than that amount, to make it worthwhile for them to reduce their output.

Germany now has such a high combination of those two kinds of generators, that there are times when their combined output is greater than domestic consumption, and that's when prices go negative.

Fourthly, although Germany shares a grid with many other continental European countries, the opportunities for spatial arbitrage are very limited, because total German generation is much much larger than the interconnection power capacity to the rest of the grid: so that interconnection capacity strictly limits the amount of energy that can be arbitraged with other countries within a given trading hour.

Finally, though it seems strange, these negative prices are a good thing. They're a temporary condition that will only last for a transition period, as electricity markets decarbonise. At the moment, we've got transmission networks and market structures that were designed for, and built around, excessively-polluting suppliers such as coal and gas. We have to stop that pollution - the cost of it way exceeds the benefits. So, the industry has to change structurally, and markets and networks will have to change too.

Coal- and gas-fuelled generators have been very helpful for in-day markets, because they can rapidly change the amount they supply. So the near-absence of participation from the consumption side hasn't been a problem at all, until recently - and why the rewards for arbitrageurs were low. But now, there is an increasing number of hours when those very flexible generators no longer dominate the market.

As we move to a grid that no longer has that supply-side responsiveness, we need responsiveness from new arbitrageurs, and from the demand-side. Period of negative pricing provide strong incentives for those new market participants to come forwards. And, as more and more of those new participants enter the market, the negative prices will become rarer and rarer.

So this isn't just Germany. It has happened, and will happen, elsewhere too.


Are there some examples of theoretical models that discuss pandemics? I am looking for some papers/references that showcase theoretical economic models that incorporate pandemics. Most work seem to be purely empirical. Does anyone know of some papers that try to model effect of pandemics on economy/economic behavior analytically?

Update:
Some of the theoretical papers that I managed to find with helpful suggestion of kurtosis to include SIR & SEIR in search include:
Daron Acemoglu et al (2020), Optimal Targeted Lockdowns In A Multi-Group SIR Model
Eichenbaum, Martin S., Sergio Rebelo, and Mathias Trabandt. The macroeconomics of epidemics.
Any further suggestions are still welcome

Many links at
https://marginalrevolution.com/?s=pandemic+model
(MR is Tyler Cohen's blog. There isn't much else to say, but my reply needs to be at least 30 characters long.)


Can vaccine distribution and appointment setting up be considered game theory? New to game theory and I started thinking about the difficulty of setting up an appointment for the covid vaccine.
Could this be something to study in game theory?
Would we need to figure out the Nash equilibrium?
The way I see it is we have N players, and each player may or may not want a vaccine, but we also have a government player who wants the most vaccinated. Each player has some payoff.

Having spent time with a few eager students of game theory myself, I sense that the "real" question you might have wanted to ask was:

What is the best way to motivate people to get inoculated, given that they have varying attitudes towards the vaccine or vaccination in general.*

To answer that question from a normative perspective, i.e. assuming everyone is rational in the economic sense, we actually need the theory not of games but of mechanism design. Game theory's primary concern is to predict the outcome(s) of some given strategic interaction, regardless of whether those outcomes are desirable (e.g. mutual defection is the predicted, albeit inefficient, outcome in Prisoner's Dilemma). On the contrary, the goal of mechanism design is to implement some desirable outcome via a strategic interaction; that is, to come up with a suitable game whose equilibrium is the outcome desired by the mechanism designer. See Dixit et al. (5th ed, ch. 14-15) for an easy introduction, or Tadelis (2013, ch.14) for a more rigorous treatment at an advanced undergraduate level.
Of particular relevance here is a special case of mechanism design called information design, where the designer (e.g. a government or public health authority) uses a suitable information revelation strategy (e.g. when and what data to publish) to persuade the public to get inoculated, given that they hold a variety of initial beliefs about the efficacy of the vaccine and will rationally update their beliefs based on information released by the designer. Information design is still at the forefront of economic research, so the linked JEL paper, which I consider above the undergraduate level, is probably as introductory a summary on this topic as you can get for now.
There is also a positive approach to address the above question. By positive I mean the  approach is founded on the acknowledgement that people are not perfectly rational and are predisposed to act in a certain way when appropriately primed. A prominent example is the nudge theory, which draws on insights from psychology and economic experiments to frame choice problems in such a way that people are more likely to pick the option favored by the so-called "choice architect" (the behavioral economic counterpart of a mechanism designer).
Lastly, although there is behavioral game theory as such, I feel that it's not as relevant to your question because much of the literature is devoted to explaining the gaps between actual play and predicted play of some canonical games.

*I'm not quite sure what you mean by "appointment setting" beyond its logistic aspect, so I will ignore it.


Interpretation of %p.a. notation Does anyone know the meaning of the notation x%p.a.?

I'm reading a text on seigniorage, and the authors have used that notation to state the usual empirical limit to the seigniorage revenues just before the inflation rates effects outweigh the money growth effect.

It's a proportional rate - 5% p.a. means 5% per annum, i.e. 5% per year


Textbooks claim that the difference between GDP and GNP (or GNI) is about geography vs citizenship--is this correct? Many textbooks claim that the difference between GDP and GNP is about geography vs citizenship. For example, Colander:

GDP is output produced within a country’s borders; GNP is output produced by a country’s citizens.

So, according to the above, if for all of 2017 through 2020, Hans the German citizen lived in the US and worked for Walmart, then his 2020 wages would count towards 2020 US GDP and German GNP (but not US GNP or German GDP).
But from my reading of the UN System of National Accounts (2008), this seems to be incorrect:

An institutional unit is said to be resident within the economic territory of a country when it maintains a centre of predominant economic interest in that territory, that is, when it engages, or intends to engage, in economic activities or transactions on a significant scale either indefinitely or over a long period of time, usually interpreted as one year.

So, by my reading, Hans should be counted as a US resident in 2020 and his 2020 Walmart wages should instead count towards US GDP.
Am I correct and are the textbooks wrong? (Or have I perhaps misinterpreted them?)

More examples of textbooks making the above claim.
Baumol and Blinder:

There is another concept, called gross national product, which counts the goods and services produced by all Americans, regardless of where they work. For consistency, the outputs produced by foreigners working in the United States are not included in GNP.

Cowen and Tabarrok:

Gross national product (GNP) is very similar to GDP but GNP measures what is produced by the labor and property supplied by U.S. permanent residents wherever in the world that labor or capital is located, rather than what is produced within the U.S. border.

Fischer and Dornbusch:

There is a distinction between GNP and gross domestic product, or GDP. GDP
is the value of final goods produced within the country. What is the difference
between GNP and GDP? Part of GNP is earned abroad. For instance, the
income of an American citizen working in Japan is part of U.S. GNP. But it is
not part of U.S. GDP because it is not earned in the United States.

O'Sullivan and Sheffrin:

The distinction between what they produce within their borders, GDP, and what their citizens earn, GNP, is not that important to most countries.

Residency is a complex definition. For example, the Council for Science and Technology of my country supports many graduate students abroad every year. Those guys stay abroad for periods of 1-5 years, yet statistics compilators still count them as residents of the National economy. Why? Because their economic interests are based at home, not the other country.
But there's more to statistics compilation than just that. Typically, the Council for Sci&Tech will provide the statistics office with data on those scholarships. But what do they do with all the folks coming and going in the corporate world? They won't report that, at least not voluntarily. So there are a few rules of thumb that the guys doing the compilation will follow:

Companies working on defined projects abroad are still residents of their home economy.
For instance, a construction company from Mexico building a bridge in Guatemala, a project that will take them 3 years, will be recorded as a resident of the Mexican economy;
Employees working for 1 year or less in another country will still be considered residents of their normal country of residence;
People whose residence moved (declared intention of moving and remaining in the new country) will now be considered residents of their new country.
Companies establishing a branch abroad will be recorded as Direct Investors, and the new branch, a resident of that economy where they are located.

Beware: this may or may not coincide with the definitions by fiscal authorities. This is strictly from a statistics compilation viewpoint. Also, these are rather guidelines, but the more information a statistical office has, the better the compilation. This is why offices of different countries should seek to cooperate with others.


Data: Vacancies in the US before 2005 Shimer (2005) says

  These [separation shocks] introduce an almost perfectly positive correlation between unemployment and vacancies, an event that essentially never been observed in the United States at business cycle frequency.

What is the data corresponding to that? Since his publication was from 2005, he must be referring to data on vacancies before the big recession. However, the standard data on job openings, JOLTS, is only available from 2005 onwards.

JOLTS actually goes back to 2000 (though years before 2005 are only available on a SIC basis). Prior to JOLTS, most people used data on the number of help-wanted ads in newspapers, such as this series from the 1950s-60s or this one that goes back to 1919.


Check if a utility function represents a monotone preference Given a function $u(x_1, x_2) = x_1 +x_2 + \min(2x_1, x_2)$, how do we mathematically prove that it monotonic or not?

Is there is a general algebraic technique to show monotonicity of suchlike functions?

Using this definition of monotone preferences (Thanks to User Art). Let $X\subseteq \mathbb{R}^n$ be the set of possible consumption bundles, and $x,y\in X$. We have weak monotone preferences if $y>>x \implies y\succ x $, and strict preferences if $y\geq x \implies y\succ x$. If we have a utility function $u: X\to \mathbb{R}$, then $y\succ x$ is the same as $u(y)> u(x)$.

Regarding notation for two bundles $x,y\in X, x=(x_1,x_2,...,x_n), y=(y_1,y_2,...,y_n)$, we say $y>>x$ if $y_i>x_i$ forall $i=1,2,...,n$ and we say $y\geq x$ if $y_i\geq x_i$ forall $i=1,2,...,n$ and there is a $j$, such that $y_j>x_j$. (This notation is not universal)

Now, see if you can use this to answer your question


Can marginal revenue be increasing? As per my calculations, the derivative of the MR curve is given by:
$MR'(q)=2p'(q)+qp''(q)$, where $p(q)$ is the inverse demand curve. Therefore,for a downward sloping demand curve, if the second derivative is large and positive, MR should actually be increasing.
However, intuitively, if demand falls, then marginal revenue should fall as well, and by a greater amount than demand. The two don't seem to match up, and that's my concern.

It is perfectly consistent for the marginal revenue to increase in $q$, even if the demand curve decreases.

Marginal revenue is
$$p(q)+ q p'(q).$$

The first term says "if I sell one extra unit then I will receive an extra $p$ in revenue". The larger is this effect, the higher is the MR.
The second term says "in order to sell one extra unit, I will have to reduce my price by $p'(q)$, which means that much less revenue on each of the $q$ units I would have sold anyway". The larger is this effect, the lower is the MR.

If $p''(q)>0$ it means the price cut needed to sell an extra unit (the second effect above) gets smaller as we move down the demand curve. So the negative part of the marginal revenue is diminished, which can cause the marginal revenue overall to increase.

Note 1: although the marginal revenue function can be increasing, it can never rise above the demand curve. Indeed, from the expression above we see that the marginal revenue is strictly below the demand curve whenever $p'(q)<0$.

Note 2 (slightly more technical): the possibility of an increasing marginal revenue implies a potential non-uniqueness of the solution to the profit-maximisation problem (even when marginal costs are constant). This is because there may be more than one point at which MR=MC, as the below figure shows (this figure was generated with the demand function $p(q)=1-q+3q^2-3q^3$):

We only have to think about points where MR=MC and MR is decreasing (the one in the figure where MR is increasing is a local minimum for profits rather than a local maximum). The other two are both local maxima, but typically only one will be the global profit-maximising point.

Becuase dealing with cases where there are multiple potential solutions is messy, people often make an assumption to guarantee a unique solution. One such assumption is that "demand is log-cocave", meaning $ln(p(q))$ is concave. Let's see why this works. Demand is log concave if

$$\frac{d^2\ln(p(q))}{dq^2}<0$$
$$\frac{p''(q)}{p(q)}-\frac{p'(q)^2}{p(q)^2}<0$$
So if demand is log-concave we know
$$p''(q)<\frac{p'(q)^2}{p(q)}.$$

We can use this information to see how large we can make the derivative of the MR curve that you calculated:
$$\frac{d MR}{dq}=2p'(q)+qp''(q)\leq 2p'(q)+q\frac{p'(q)^2}{p(q)}.$$
Can this be positive? The answer is yes if
$$2p'(q)+q\frac{p'(q)^2}{p(q)}>0$$
i.e. if
$$2p(q)+qp'(q)=MR+p(q)<0.$$
(in the last line I changed the direction of the inequality because I divided by $p'$, which is negative).

We find that, given log-concave demand, $\frac{d MR}{dq}$ can be positive only if $MR<0$. So the marginal revenue curve will never slope up in the region the firm would consider choosing.


Under what condition would the law of one price hold? Under the consumption-based model for asset pricing, different people will have different prices because of their different utility functions.

What is the force that make the law of one price hold?

You might say that it is a result of no arbitrage, but Kerry Back states in his book Asset Pricing and Portfolio Choice Theory states the following

  The law of one price is a weaker condition than absence of arbitrage opportunities: It is implied by the absence of arbitrage opportunities, but it does not imply the absence of arbitrage opportunities.

Based on this statement, absence of arbitrage is a stronger condition; the law of one price should hold even there is arbitrage in the market.
So What is the force that make the law of one price hold?

Converting my comments to an answer. The law of one price (LoP) is an economic concept which posits that "a good must sell for the same price in all locations". This law is derived from the assumption of the inevitable elimination of all arbitrage. Source wikipedia.

Arbitrage opportunities imply the law of one price but the reverse is not always true. In logic, LoP is a necessary condition for arbitrage but not a sufficient one. In common language, we cannot out rule out the fact that the law of one price may hold without full arbitrage. If the price of an asset is not different for different people, the law of one price holds without assuming full arbitrage.


Secular Stagnation: A Plain-English Explanation Put aside the mathematics and jargon for a couple minutes. If you had to explain the concept of secular stagnation to an (intelligent) person with no economics background, how would you go about doing so?

If you want to use terms like "zero lower bound" or "tight money" ... explain them thoroughly!

Quoting from Coen Teulings C and
Richard Baldwin R (ed.) (2014) Secular Stagnation:
Facts, Causes and Cures

  "(...) A workable definition of secular stagnation is that negative real interest rates are needed to equate saving and investment with
  full employment. (p.2)"

Later on they write (p.10)

  "Low real interest rates play a linchpin role in the secular stagnation debate for two reasons.
  • First, if real rates are
  low in normal times, adverse macroeconomic shocks are more likely to
  require negative real rates to restore a full-employment
  investment-savings balance. In today’s low-inflation environment, this
  tends to undermine the effectiveness of monetary policy.
  •
  Second, low nominal and real interest rates undermine financial stability.   Summers (i.e. Larry Summers, the economist that brought
  back the concept of Secular Stagnation) lists three channels through
  which low rates may foster instability:
  (i) they increase risk-taking as investors reach for yield;
  (ii) they promote irresponsible lending as coupon obligations become very low and easy to meet; and
  (iii) they make Ponzi financial structures more attractive as interest rates look low relative to expected growth rates."

So if interest rates tend to be low, the need to push them down even more in order to deal with macroeconomic shocks, may lead to financially-born crises.

So why interest rates have historically fallen? From the supply side (i.e. savings behavior/supply of loans) factors like,
a) increase in life expectancy
b) lower retirement age, and
c) the decline in the growth rate of the population,
tend to increase required life time savings - required in order to smooth life-time consumption (at the individual level this goes like this: projecting that we will live more, and stay in pension for more years, we tend to save more during our working years).
I will add that supply of loans will tend to increase with wealth inequality.

The effect from the demand side (i.e. investment behavior/demand for loans) is much less agreed upon, but it appears that the revolution of the Information Technology (IT) sector had a negative effect on this aspect, since it reduced the value of the investments needed. The authors write

  "The key is that the development of high value-added services by Google, Microsoft, Amazon, Facebook and the like require relatively
  little investment.

So the loan supply schedule shifted outwards, while the loan demand scheduled shifted inwards. Schematically.

which creates the needed environment for Secular Stagnation, if it goes too far down.

Of course this is not the whole story. The above e-book by CEPR is freely downloadable.


Would the market stabilize if companies paid taxes on revenue instead of gross income? Note: there is a closed question that is similar (https://money.stackexchange.com/questions/126677/paying-taxes-revenue-vs-taxes-profit) but the response offers a perspective that I would like to challenge and understand.
Companies pay taxes on gross income (revenue - cost of sold goods) while individuals pay directly on revenue.
Several sources I could read (including the answer to the question I mentioned) mention fairness between companies and the fact that they "spend to make money" and not "consume" as in the case of an individual (why would one be better than the other?)
Not being an economist, I still do not understand why the difference in treatment.
If taxes were paid on revenue, wouldn't the money flow stabilize somewhere else? (prices would go up, and so would salaries in order for people to afford the new prices).
One of the advantages I imagine would be that countries that have lower taxation would not be artificially used as a consolidation center to show a "zero (or small) gross income" (and therefore low or zero taxes) in other countries.
To be clear: I am not trying to be the Robin Hood of economics, I am trying to understand whether there are fundamental reasons why the market would not stabilize elsewhere.

If taxes were paid on revenue, wouldn't the money flow stabilize somewhere else? (prices would go up, and so would salaries in order for people to afford the new prices).

This sentence is unclear. If you mean to ask whether market would eventually reach some stable equilibrium the answer would be yes.
However, this is a bit of a moot point. There actually already are taxes on revenue as pointed in comments. VAT and sales taxes are de facto taxes on company's revenue even if they might not be de jure taxes on company revenue. This is because company's revenue is defined as price times quantity sold $R=pq$ so any tax that is levied on price company charges is de facto a revenue tax.
I make a purposeful distinction between de facto and de jure tax because in most tax codes you will see written that firms collect VAT/sales tax from consumers on behalf of the government. So de jure these taxes are applied to consumers, however from economic perspective it does not matter if VAT or sales tax is de jure levied on consumers or producers as tax burden is always allocated by the market (see Mankiw Principles of Economics).
The intuition why they are equivalent can be seen in the pictures below I taken from the Mankiw's book. The intuition is that whether tax is levied on consumers or suppliers (firms) it will have the same effect on new market equilibrium price and quantity supplied.

So de facto VAT and sales taxes are also revenue taxes. Also since tax burden is split by the market it does not really matter from distributional/welfare perspective if government puts this tax de jure on businesses, so there is no opportunity here to do any redistribution just by changing the de jure designated payer of a tax. This does not mean the burden of the tax is distributed equally or that even there might not be cases where only one side of the market bears the burden. The point is that the burden is determined by the market not government.
In practice the designation of who pays might have political significance. That is an economically equivalent tax on producers might be more palatable to voters to the same tax on consumers but that does not matter for economic analysis.
Furthermore, an important caveat is that sometimes it might be more practical to tax one party because they might have for example lower administrative burden. This is why VAT and sales tax are both collected by companies so in fact they pay the tax even though de jure these are taxes on consumers and de facto the tax burden is always split between consumers and producers depending on parameters of supply and demand regardless of designated payer stipulated in law.


What is a good way te represent analytically two markets with different price elasticities of demand? It seems that two linear demand curves with different slopes would be a good way to do it, but linear demand curves are hard to work with otherwise.

How about constant elasticity demand: $Q(p)=p^a$, where $a<0$?

Such functions, as their name implies, have the advantage that the price elasticity of demand is constant along their entire length:

$$\eta=Q'(p)\frac{p}{q}=ap^{a-1}\frac{p}{p^a}=a.$$

Since the elasticity of the demand function is directly controlled with a single parameter, such functions are often convenient when performing comparative statics exercises in which the objective is to show the effect of varying the elasticity of demand upon equilibrium.


How can 1 BTC gain value in USD if inflation is happening? So, right now, 1 BTC (bitcoin) is worth ≈ 578.25 USD. But, with the inflation of the USD, how can 1 BTC still hold its value at 578.25? Why doesn't it decrease?

Let's set this example:

1 BTC = 1 USD.

Then, the USD loses half of its value; 1 USD is now actually worth 0.50.

Why won't the BTC lose half of its value?

I guess this principle applies to any currency and I can't seem to grasp the idea.

So, any help will be appreciate.

I also hope that you can understand my question and if you need clarification, please ask!

I'll try to fit this in an answer. I think you are confusing nominal and relative value. Inflation means that the 'value' of your money depreciates relative to other things. For example, in the year 2000 I could buy a bread for 1 USD. In the year 2015 I can buy a bread for 2 USD. This means that the dollar lost half of it's value relative to bread. But nominal 1 USD is still 1 USD and 1 bread is still 1 bread.

So in your example: today 1 BTC = 578.25 USD. If there is inflation of the USD (the value depreciates), then all else being equal BTC should appreciate against the USD (go up in nominal USD-value). So to work further with the bread-example. If in the year 2000 1 BTC would buy 500 USD then in 2015 1 BTC should buy 1000 USD. So the USD went down in value relative to BTC.

for more information:
https://en.wikipedia.org/wiki/Purchasing_power
http://www.investopedia.com/terms/c/currency-depreciation.asp


Floating exchange rate and cost-push inflation? I have a trouble understanding the following text from my textbook. I bolded the fragments that don't make sense to me.

"Disadvantages of a floating exchange rate:
A floating exchange rate regime may worsen existing levels of inflation. If a country has high inflation relative to other countries then this will make its exports less competitive and its imports relatively less expensive. The exchange rate will then fall, in order to rectify the situation. However, this could lead to even higher import prices of finished goods, componetns, and raw materials, and thus cost-push inflation, which may further fuel the overall inflation rate."

If there's high inflation, exports become less competitive and imports become less expensive - that's obvious. But why do they say that it would make imports' prices even higher? Why "even" and why "higher"? They've said that imports are cheaper after inflation, so imported raw materials, etc. should be cheaper as well so the costs of productions should be cheaper too and don't cause cost-push inflation but totally opposite.

I would imagine that a falling exchange rate causes the price of imports to rise as the foreign currency becomes stronger against the local currency.

With high inflation exports become more expensive because foreign buyers have to exchange more of their currency to account for the increase in price. This means they cost more per unit in terms of the foreign currency and consequently become less competitive.

The subsequent fall in demand for exports causes the fall in the exchange rate as the demand for exports and local currency are complementary. A weaker local currency means that one unit of local currency is now worth fewer units of foreign currency.

Thus the result is that imports become more expensive as any given amount of imports now costs more in terms of the local currency. Imagine that the exporting nation, say England, sells teddy bears for £5.00 a piece. If the dollar weakened against the pound from \$1.50 to a pound to \$1.56, then as far as a buyer from the United States is concerned, the price of a teddy bear increases from \$7.5 (i.e. 1.50 * 5.00) to \$7.8 (i.e. 1.56 * 5.00).


Why are shares exempt from GST? Why is the financial trading sector exempted from consumption taxes?
For example, shares seem to be exempt from GST in both Australia and Canada. What is the intention behind this? Is it common for jurisdictions to have a VAT or sales tax that also covers or excludes financial instruments and supplies?
Semantics aside, is there something that would make it infeasible for some kind of sales tax to be levied at an identical rate on every transaction in the economy?

Shares, and many other financial instruments as well, are financial assets  assets, through which people save not consumption of goods and services. Consequently, neither GST (goods and services tax) or value added tax (which is in economics a type of consumption tax) can apply to stocks or many other financial instruments as they do not represent consumption of either goods nor services. Hence prima facie there is no reason why consumption taxes should apply to things that are neither consumption or goods or services in the first place. That is like asking why houses are exempt from labor taxes... well its because they are capital not labor.
For example, if you pay someone to make trades for you (and hence someone to provide you financial service), you will generally be required to pay VAT or GST on that (unless particular country decided to exempt it for whatever political or legal reason - which is beyond the professional competency of an economist to comment on).

Response to edit:
It is feasible to tax financial assets but this would be considered wealth taxation (completely different type of taxation than consumption taxes). Taxing wealth is feasible in a sense government can pass legislation to actually do it, but from economic perspective it is often considered undesirable for its negative impact on the economy.
The literature shows that optimal top marginal wealth taxes, even in case of Rawlsian social utility function (the most redistributive social function there is), are somewhere in range 0-10% (see Kocherlakota 2005; Fama 2019; Zucman and Saez 2019 or this article by Summers) or discussion in the famous Mirrlees Review: Dimensions of Tax Design). Also, note the consensus seems to still be closer to the zero result and only very few scholars would go as high as 10%.
The above being said governments typically do not conduct science based policies. Often politicians will ignore even recommendation of their own analysts when it comes to the design of tax policy. Consequently, it is beyond scope of economic profession to tell you why politicians do not base all their laws on best published science/state of the art models or why they do tax one thing but not another (for example, in the Netherlands where I live for long time food for hamsters and bunnies was taxed at different rate even though they are perfect substitutes - which is completely absurd from any economic perspective, yet that is what the government legislated).